residual optimal transport

This work addresses scalability and stability challenges in computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces. We propose an end-to-end framework based on Conditional Normalizing Flows (CNFs), which jointly maps multiple source distributions into a shared latent space via invertible implicit transport—directly modeling the transport maps and enabling closed-form barycenter computation, thereby circumventing conventional dual optimization and adversarial training. To our knowledge, this is the first method to employ CNFs for joint learning of multi-source transport maps, supporting efficient barycenter estimation across up to hundreds of input distributions. Experiments demonstrate that our approach significantly outperforms existing state-of-the-art methods on high-dimensional tasks, achieving substantial improvements in accuracy, computational efficiency, and training stability.

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.

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.

This work addresses the challenge of generative modeling on measure spaces, particularly the transport of measures over measures (metameasures), by introducing the Wasserstein-on-Wasserstein (WoW) framework. It extends flow matching to the Wasserstein space of probability measures for the first time, leveraging a nested optimal transport plan—comprising both outer and inner couplings—to naturally induce a velocity field and define a deterministic dynamical system. By integrating nested Wasserstein geometry, sliced Wasserstein distances, and linearized approximations, the method achieves substantially improved computational efficiency and numerical stability. The resulting generative trajectories are nearly straight in the Wasserstein sense, yielding state-of-the-art performance with strong theoretical coherence in point cloud and set generation tasks.

To address the high computational cost and curse-of-dimensionality challenges in sampling optimal transport (OT) couplings for large-scale, high-dimensional data, this paper proposes an efficient learning framework based on score-based generative models. Specifically, conditioned on source samples, it iteratively generates target samples following the Sinkhorn-regularized OT coupling via Langevin dynamics. Crucially, it jointly parameterizes the score function and Sinkhorn potential functions—enabling, for the first time, end-to-end co-learning of score-based generation and OT coupling. We theoretically establish the convergence of gradient descent on the network parameters under mild assumptions. Experiments demonstrate that our method significantly improves both accuracy and speed of coupling estimation across diverse large-scale OT tasks, while maintaining scalability and practical applicability.

This work proposes a continuous normalizing flow method based on the generalized Benamou–Brenier formulation to solve general $p$-cost optimal transport ($p$-OT) problems. The approach parameterizes the velocity field via the gradient of a scalar potential function and introduces a self-induced matching loss, training the model along the straight-line bridge defined by its own endpoints while leveraging maximum mean discrepancy (MMD) for flexible terminal distribution matching. Under regularity, exact terminal matching, and uniqueness assumptions, the authors theoretically establish that zero-loss solutions strictly satisfy the generalized Benamou–Brenier optimality system, thereby exactly recovering the $p$-OT map and dynamics. Experiments demonstrate that the method accurately reconstructs theoretical $p$-OT maps on synthetic data, achieves strong performance in high-dimensional tabular density modeling, and validates the flexibility of terminal matching in sample-wise color transfer tasks.

This work systematically addresses statistical uncertainty in probability measures within the optimal transport framework. By extending classical optimal transport theory to the setting of random probability measures, it establishes—for the first time—the Riemannian geometric structure of the L²-type Wasserstein space, characterizing its metric and geodesics, and introduces a stochastic flow whose sample paths follow Wasserstein gradient flows. This approach unifies statistical inference and generative modeling, extends Schwartz’s posterior consistency theorem to the Wasserstein topology, and provides convergence guarantees for principled inference under random sampling. Furthermore, it offers a theoretical foundation compatible with stochastic token sampling in Transformer architectures.

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.

This work addresses the limitations of traditional conditional optimal transport (COT) in conditional generative modeling, which is sensitive to outliers and constrained by rigid distribution-matching requirements when data subsets are limited. The authors propose a conditional unbalanced optimal transport (CUOT) framework, introducing unbalanced optimal transport into conditional generation for the first time. By relaxing the conditional distribution matching constraint via Csiszár divergences while strictly preserving the conditional marginal distributions, the method incorporates a triangular map parameterization that satisfies the c-transform relationship. Built upon semi-dual optimization, the resulting CUOT model enjoys both theoretical guarantees and enhanced robustness. Experiments demonstrate that CUOT significantly outperforms existing COT approaches on both 2D synthetic and image datasets, exhibiting superior performance in outlier robustness, distribution matching accuracy, and sampling efficiency.

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.