optimal transport

This work addresses the problem of probabilistic distribution modeling and robust classification for point cloud data under few-shot learning settings. We propose a novel framework unifying probabilistic measure synthesis (via weighted Wasserstein-2 barycenters) and analysis (via barycentric coordinate estimation), regularized by entropy. For the first time under minimal assumptions, we derive its gradient analytically and construct a convex quadratic programming solver based on the entropic map fixed-point equation. We establish dimension-free convergence rates for barycentric coordinates and Wasserstein stability guarantees. Integrating Sinkhorn divergence, entropic maps, and optimal transport, our method significantly improves recognition accuracy for corrupted point cloud classes in few-shot scenarios—outperforming neural network baselines. Extensive experiments validate both theoretical convergence and robustness to noise.

This work addresses the challenge of defining weighted averages (barycenters) of probability measures on graph structures, where classical optimal transport suffers from geometric degeneracy. By leveraging the Riemannian geometry of the probability simplex induced by dynamic optimal transport, the authors propose an intrinsic gradient descent method to compute barycenters on graphs. This approach approximates the exponential map and its inverse through numerical geodesic approximation and discretization of the continuity equation. In the analysis phase, barycentric coordinates are obtained by solving a quadratic program based on geodesic distances. Experimental results demonstrate that the proposed method significantly outperforms static optimal transport approaches relying on graph distances or entropic regularization, thereby validating the effectiveness and superiority of the introduced intrinsic geometric framework for measure synthesis and analysis on graphs.

This paper addresses statistical inference for optimal transport (OT) maps by developing a nonparametric estimation framework and asymptotic theory grounded in empirical data. Methodologically, it integrates optimal transport theory, convergence analysis of probability measures, and large-sample statistical techniques to rigorously characterize estimation consistency and limiting distributions for diverse OT maps—including continuous, discrete, and regularized variants. The work introduces a unified inferential paradigm, providing the first rigorous asymptotic guarantees for point estimation, confidence band construction, and hypothesis testing of OT maps. These theoretical advances substantially enhance the interpretability and reliability of OT in practical applications such as causal inference, generative modeling, and distribution alignment. The resulting statistical toolkit—comprising estimators, uncertainty quantification procedures, and operational guidelines—is designed for cross-domain deployment and reproducible implementation. (149 words)

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 paper addresses the regularized estimation of multivariate quantiles in Banach spaces, focusing on efficient computation of entropy-regularized optimal transport (EOT) within the Monge–Kantorovich quantile framework. We propose a novel stochastic EOT algorithm: when the source measure μ is uniform over the unit hypercube or sphere, we parameterize the Kantorovich dual potentials using Fourier bases—marking the first integration of Fourier representation with stochastic EOT. Theoretically, we establish almost-sure convergence of the algorithm in infinite-dimensional Banach spaces. Computationally, each iteration requires only two fast Fourier transforms (FFTs), yielding substantial efficiency gains. Our approach endows entropy regularization with an explicit interpretation as smooth quantile modeling. Empirical evaluations on synthetic and real-world datasets demonstrate superior robustness and accuracy—particularly for small-sample quantile estimation.

This study addresses the absence of systematic guidance on optimal transport theory and its applications in econometrics. It proposes a novel classification of optimal transport methods grounded in their mathematical structure and constructs a coherent theoretical framework that precisely maps distinct optimal transport approaches to specific econometric contexts. By integrating optimal transport theory, econometric modeling, and advanced mathematical analysis, this work establishes an accessible and actionable methodological bridge for both theoretical and applied econometricians. The framework substantially enhances the applicability and accessibility of optimal transport in empirical and structural econometric models, facilitating its broader adoption and rigorous implementation in economic research.

This study addresses the two-sample homogeneity testing problem by proposing a novel approach based on entropy-regularized optimal transport (EOT) maps. The test statistic is constructed as the squared L² distance between two empirical EOT maps evaluated over the unit sphere, and its non-pivotal null distribution is calibrated via a weighted multiplier bootstrap. This work is the first to incorporate EOT maps into two-sample testing, enabling not only detection of global distributional discrepancies but also diagnosis of their specific patterns. Theoretical analysis establishes a Gaussian quadratic form as the limiting null distribution and derives asymptotic power under local alternatives. Numerical experiments demonstrate that the method achieves accurate size control and high power in finite samples, exhibiting particular sensitivity to location shifts, while both simulations and real-data analyses corroborate its effectiveness and diagnostic capability.

The Knothe–Rosenblatt (KR) rearrangement lacks a variational construction framework, hindering its integration into optimal transport (OT) theory. Method: We propose a novel approximation scheme based on soft-constrained OT: a weighted quadratic-cost relaxed optimization problem constructs the KR map dimension-by-dimension while satisfying marginal constraints. We extend this to dynamic OT, designing triangular-structured velocity fields that unify static mapping with continuous-time dynamical modeling. The method jointly minimizes pushforward measure divergence and transforms conditional distributions. Contribution/Results: We provide the first rigorous proof that the constructed KR map converges to the solution of the soft-constrained OT problem in the limit. The framework ensures theoretical soundness while enhancing flexibility in conditional sampling and mapping estimation. Experiments demonstrate its effectiveness for high-dimensional KR map estimation, offering a new variational tool bridging OT theory and practical applications.

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 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.