Optimal Transport

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Optimal transport studies how probability mass can be moved between distributions at minimum cost, giving a geometric way to compare data even when samples do not correspond one-to-one. Wasserstein distances and transport maps describe this geometry, while Gromov–Wasserstein methods compare relational structure when data live in different spaces. Practical approaches include entropic regularization with Sinkhorn algorithms, sliced and low-rank approximations, and neural transport maps used in generative modeling, domain adaptation, and trajectory inference. A central challenge is balancing computational and statistical scalability with faithful transport, especially when choosing regularization or relaxing exact mass constraints.

Latest papers

289 papers48 papers in the last four weeks, up 17% on the four weeks before. 0.3% of all new papers.

Apr 3, 2026

Conditional Sampling via Wasserstein Autoencoders and Triangular Transport

We present Conditional Wasserstein Autoencoders (CWAEs), a framework for conditional simulation that exploits low-dimensional structure in both the conditioned and the conditioning variables. The key idea is to modify a Wasserstein autoencoder to use a (block-) triangular decoder and impose an appropriate independence assumption on the latent variables. We show that the resulting model gives an autoencoder that can exploit low-dimensional structure while simultaneously the decoder can be used for conditional simulation. We explore various theoretical properties of CWAEs, including their connections to conditional optimal transport (OT) problems. We also present alternative formulations that lead to three architectural variants forming the foundation of our algorithms. We present a series of numerical experiments that demonstrate that our different CWAE variants achieve substantial reductions in approximation error relative to the low-rank ensemble Kalman filter (LREnKF), particularly in problems where the support of the conditional measures is truly low-dimensional.

Mohammad Al-Jarrah, Michele Martino, Marcus Yim +2

Department of Aeronautics & Astronautics, University of Washington, Seattle · Department of Applied Mathematics, University of Washington, Seattle · Department of Mathematics, University of Washington, Seattle

Conditional Generative Modeling Optimal Transport

Apr 3, 2026

Structure-Preserving Multi-View Embedding Using Gromov-Wasserstein Optimal Transport

Multi-view data analysis seeks to integrate multiple representations of the same samples in order to recover a coherent low-dimensional structure. Classical approaches often rely on feature concatenation or explicit alignment assumptions, which become restrictive under heterogeneous geometries or nonlinear distortions. In this work, we propose two geometry-aware multi-view embedding strategies grounded in Gromov-Wasserstein (GW) optimal transport. The first, termed Mean-GWMDS, aggregates view-specific relational information by averaging distance matrices and applying GW-based multidimensional scaling to obtain a representative embedding. The second strategy, referred to as Multi-GWMDS, adopts a selection-based paradigm in which multiple geometry-consistent candidate embeddings are generated via GW-based alignment and a representative embedding is selected. Experiments on synthetic manifolds and real-world datasets show that the proposed methods effectively preserve intrinsic relational structure across views. These results highlight GW-based approaches as a flexible and principled framework for multi-view representation learning.

Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Canindé, Ceará, Brazil · Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, Ceará, Brazil · Sigma Nova, Paris, France

Multidimensional Scaling Multi-View Learning Dimensionality Reduction Multi-View Geometry Optimal Transport

Apr 2, 2026

HOT: Harmonic-Constrained Optimal Transport for Remote Photoplethysmography Domain Adaptation

Remote photoplethysmography (rPPG) enables non-contact physiological measurement from facial videos; however, its practical deployment is often hindered by substantial performance degradation under domain shift. While recent deep learning-based rPPG methods have achieved strong performance on individual datasets, they frequently overfit to appearance-related factors, such as illumination, camera characteristics, and color response, that vary significantly across domains. To address this limitation, we introduce frequency domain adaptation (FDA) as a principled strategy for modeling appearance variation in rPPG. By transferring low-frequency spectral components that encode domain-dependent appearance characteristics, FDA encourages rPPG models to learn invariance to appearance variations while retaining cardiac-induced signals. To further support physiologically consistent alignment under such appearance variation, we propose Harmonic-Constrained Optimal Transport (HOT), which leverages the harmonic property of cardiac signals to guide alignment between original and FDA-transferred representations. Extensive cross-dataset experiments demonstrate that the proposed FDA and HOT framework effectively enhances the robustness and generalization of rPPG models across diverse datasets.

Ba-Thinh Nguyen, Thi-Duyen Ngo, Thanh-Trung Huynh +2

VNU University of Engineering and Technology, Hanoi, Vietnam · College of Engineering and Computer Science, VinUniversity, Hanoi, Vietnam · VinUni-Illinois Smart Health Center, VinUniversity, Hanoi, Vietnam +1

Remote Photoplethysmography Optimal Transport Frequency-Domain Feature Learning

Apr 1, 2026

Causal Optimal Coupling for Gaussian Input-Output Distributional Data

We study the problem of identifying an optimal coupling between input-output distributional data generated by a causal dynamical system. The coupling is required to satisfy prescribed marginal distributions and a causality constraint reflecting the temporal structure of the system. We formulate this problem as a Schr"odinger Bridge, which seeks the coupling closest - in Kullback-Leibler divergence - to a given prior while enforcing both marginal and causality constraints. For the case of Gaussian marginals and general time-dependent quadratic cost functions, we derive a fully tractable characterization of the Sinkhorn iterations that converges to the optimal solution. Beyond its theoretical contribution, the proposed framework provides a principled foundation for applying causal optimal transport methods to system identification from distributional data.

Daran Xu, Amirhossein Taghvaei

Department of Applied Mathematics, University of Washington, Seattle, WA · Department of Aeronautics and Astronautics, University of Washington, Seattle, WA

Optimal Transport Schrödinger Bridges

Mar 30, 2026

An Explicit Surrogate for Gaussian Mixture Flow Matching with Wasserstein Gap Bounds

We study training-free flow matching between two Gaussian mixture models (GMMs) using explicit velocity fields that transport one mixture into the other over time. Our baseline approach constructs component-wise Gaussian paths with affine velocity fields satisfying the continuity equation, which yields to a closed-form surrogate for the pairwise kinetic transport cost. In contrast to the exact Gaussian Wasserstein cost, which relies on matrix square-root computations, the surrogate admits a simple analytic expression derived from the kinetic energy of the induced flow. We then analyze how closely this surrogate approximates the exact cost. We prove second-order agreement in a local commuting regime and derive an explicit cubic error bound in the local commuting regime. To handle nonlocal regimes, we introduce a path-splitting strategy that localizes the covariance evolution and enables piecewise application of the bound. We finally compare the surrogate with an exact construction based on the Gaussian Wasserstein geodesic and summarize the results in a practical regime map showing when the surrogate is accurate and the exact method is preferable.

Elham Rostami, Taous-Meriem Laleg-Kirati, Hamidou Tembine

Université Paris-Saclay, Inria, CIAMS, Gif-sur-Yvette, 91190, France · Department of Electrical Engineering and Computer Science, University of Quebec in Trois-Rivieres (UQTR), Quebec, Canada

Flow Matching Optimal Transport Surrogate Modeling Wasserstein Distance Gaussian Mixture Models

Mar 29, 2026

Statistical Guarantees for Distributionally Robust Optimization with Optimal Transport and OT-Regularized Divergences

We study finite-sample statistical performance guarantees for distributionally robust optimization (DRO) with optimal transport (OT) and OT-regularized divergence model neighborhoods. Specifically, we derive concentration inequalities for supervised learning via DRO-based adversarial training, as commonly employed to enhance the adversarial robustness of machine learning models. Our results apply to a wide range of OT cost functions, beyond the

p

-Wasserstein case studied by previous authors. In particular, our results are the first to: 1) cover soft-constraint norm-ball OT cost functions; soft-constraint costs have been shown empirically to enhance robustness when used in adversarial training, 2) apply to the combination of adversarial sample generation and adversarial reweighting that is induced by using OT-regularized

f

-divergence model neighborhoods; the added reweighting mechanism has also been shown empirically to further improve performance. In addition, even in the

p

-Wasserstein case, our bounds exhibit better behavior as a function of the DRO neighborhood size than previous results when applied to the adversarial setting.

Jeremiah Birrell, Xiaoxi Shen

Department of Mathematics Texas State University San Marcos, TX, USA

Distributionally Robust Optimization Wasserstein Distributionally Robust Optimization Statistical Learning Theory Adversarial Training Optimal Transport

Mar 27, 2026

Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs

The optimal transportation problem defines a geometry of probability measures which leads to a definition for weighted averages (barycenters) of measures, finding application in the machine learning and computer vision communities as a signal processing tool. Here, we implement a barycentric coding model for measures which are supported on a graph, a context in which the classical optimal transport geometry becomes degenerate, by leveraging a Riemannian structure on the simplex induced by a dynamic formulation of the optimal transport problem. We approximate the exponential mapping associated to the Riemannian structure, as well as its inverse, by utilizing past approaches which compute action minimizing curves in order to numerically approximate transport distances for measures supported on discrete spaces. Intrinsic gradient descent is then used to synthesize barycenters, wherein gradients of a variance functional are computed by approximating geodesic curves between the current iterate and the reference measures; iterates are then pushed forward via a discretization of the continuity equation. Analysis of measures with respect to given dictionary of references is performed by solving a quadratic program formed by computing geodesics between target and reference measures. We compare our novel approach to one based on entropic regularization of the static formulation of the optimal transport problem where the graph structure is encoded via graph distance functions, we present numerical experiments validating our approach, and we conclude that intrinsic gradient descent on the probability simplex provides a coherent framework for the synthesis and analysis of measures supported on graphs.

David Gentile, James M. Murphy

Tufts University, Dept. of Mathematics, Medford, MA, USA

Wasserstein Barycenters Optimal Transport

Mar 24, 2026

Wasserstein Parallel Transport for Predicting the Dynamics of Statistical Systems

Many scientific systems, such as cellular populations or economic cohorts, are naturally described by probability distributions that evolve over time. Predicting how such a system would have evolved under different forces or initial conditions is fundamental to causal inference, domain adaptation, and counterfactual prediction. However, the space of distributions often lacks the vector space structure on which classical methods rely. To address this, we introduce a general notion of parallel dynamics at a distributional level. We base this principle on parallel transport of tangent dynamics along optimal transport geodesics and call it ``Wasserstein Parallel Trends''. By replacing the vector subtraction of classic methods with geodesic parallel transport, we can provide counterfactual comparisons of distributional dynamics in applications such as causal inference, domain adaptation, and batch-effect correction in experimental settings. The main mathematical contribution is a novel notion of fanning scheme on the Wasserstein manifold that allows us to efficiently approximate parallel transport along geodesics while also providing the first theoretical guarantees for parallel transport in the Wasserstein space. We also show that Wasserstein Parallel Trends recovers the classic parallel trends assumption for averages as a special case and derive closed-form parallel transport for Gaussian measures. We deploy the method on synthetic data and two single-cell RNA sequencing datasets to impute gene-expression dynamics across biological systems.

Tristan Luca Saidi, Gonzalo Mena, Larry Wasserman +1

Department of Statistics and Data Science, Carnegie Mellon University · Machine Learning Department, Carnegie Mellon University · Department of Economics, Emory University

Batch Effect Correction Causal Counterfactual Generation Optimal Transport Causal Inference

Mar 21, 2026

Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics

We propose a dense associative memory for empirical measures (weighted point clouds). Stored patterns and queries are finitely supported probability measures, and retrieval is defined by minimizing a Hopfield-style log-sum-exp energy built from the debiased Sinkhorn divergence. We derive retrieval dynamics as a spherical Hellinger Kantorovich (SHK) gradient flow, which updates both support locations and weights. Discretizing the flow yields a deterministic algorithm that uses Sinkhorn potentials to compute barycentric transport steps and a multiplicative simplex reweighting. Under local separation and PL-type conditions we prove basin invariance, geometric convergence to a local minimizer, and a bound showing the minimizer remains close to the corresponding stored pattern. Under a random pattern model, we further show that these Sinkhorn basins are disjoint with high probability, implying exponential capacity in the ambient dimension. Experiments on synthetic Gaussian point-cloud memories demonstrate robust recovery from perturbed queries versus a Euclidean Hopfield-type baseline.

Aratrika Mustafi, Soumya Mukherjee

Department of Statistics, Pennsylvania State University

Sinkhorn Divergence Associative Memory Optimal Transport

Mar 17, 2026

SuCor: Susceptibility Distortion Correction via Parameter-Free and Self-Regularized Optimal Transport

We present SuCor, a method for correcting susceptibility induced geometric distortions in echo planar imaging (EPI) using optimal transport (OT) along the phase encoding direction. Given a pair of reversed phase encoding EPI volumes, we model each column of the distortion field as a Wasserstein-2 barycentric displacement between the opposing-polarity intensity profiles. Regularization is performed in the spectral domain using a bending-energy penalty whose strength is selected automatically via the Morozov discrepancy principle, requiring no manual tuning. On a human connectome project (HCP) dataset with left-right/right-left b0 EPI pairs and a co-registered T1 structural reference, SuCor achieves a mean volumetric mutual information of 0.341 with the T1 image, compared to 0.317 for FSL TOPUP, while running in approximately 12 seconds on a single CPU core.

Sreekar Chigurupati, Eleftherios Garyfallidis

Intelligent Systems Engineering, Indiana University, Bloomington, Indiana, USA.

Magnetic Resonance Imaging Neuroimaging Optimal Transport

Mar 16, 2026

Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport

We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}. Persistence spheres map an integrable measure

μ

on the upper half-plane, including persistence diagrams (PDs) as counting measures, to a function

S(μ)\in C(\mathbb{S}^2)

, and the map is stable with respect to 1-Wasserstein partial transport distance

\mathrm{POT}_1

. Moreover, to the best of our knowledge, persistence spheres are the first explicit representation used in topological machine learning for which continuity of the inverse on the image is established at every compactly supported target. Recent bounded-cardinality bi-Lipschitz embedding results in partial transport spaces, despite being powerful, are not given by the kind of explicit summary map considered here. Our construction is rooted in convex geometry: for positive measures, the defining ReLU integral is the support function of the lift zonoid. Building on~\cite{pegoraro2025persistence}, we refine the definition to better match the

\mathrm{POT}_1

deletion mechanism, encoding partial transport via a signed diagonal augmentation. In particular, for integrable

μ

, the uniform norm between

S(0)

and

S(μ)

depends only on the persistence of

μ

, without any need of ad-hoc re-weightings, reflecting optimal transport to the diagonal at persistence cost. This yields a parameter-free representation at the level of measures (up to numerical discretization), while accommodating future extensions where

μ

is a smoothed measure derived from PDs (e.g., persistence intensity functions~\citep{wu2024estimation}). Across clustering, regression, and classification tasks involving functional data, time series, graphs, meshes, and point clouds, the updated persistence spheres are competitive and often improve upon persistence images, persistence landscapes, persistence splines, and sliced Wasserstein kernel baselines.

Matteo Pegoraro

Institute of Computing, Faculty of Informatics, Università della Svizzera Italiana

Topological Representation Learning Persistent Homology Optimal Transport

Mar 16, 2026

Faster Inference of Flow-Based Generative Models via Improved Data-Noise Coupling

Conditional Flow Matching (CFM), a simulation-free method for training continuous normalizing flows, provides an efficient alternative to diffusion models for key tasks like image and video generation. The performance of CFM in solving these tasks depends on the way data is coupled with noise. A recent approach uses minibatch optimal transport (OT) to reassign noise-data pairs in each training step to streamline sampling trajectories and thus accelerate inference. However, its optimization is restricted to individual minibatches, limiting its effectiveness on large datasets. To address this shortcoming, we introduce LOOM-CFM (Looking Out Of Minibatch-CFM), a novel method to extend the scope of minibatch OT by preserving and optimizing these assignments across minibatches over training time. Our approach demonstrates consistent improvements in the sampling speed-quality trade-off across multiple datasets. LOOM-CFM also enhances distillation initialization and supports high-resolution synthesis in latent space training.

Aram Davtyan, Leello Tadesse Dadi, Volkan Cevher +1

University of Bern, Bern, Switzerland · EPFL, Lausanne, Switzerland

Flow-Based Generative Modeling Conditional Flow Matching Optimal Transport

Mar 15, 2026

Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport

We introduce a framework for analyzing topological tipping in time-evolutionary point clouds by extending the recently proposed Topological Optimal Transport (TpOT) distance. While TpOT unifies geometric, homological, and higher-order relations into one metric, its global scalar distance can obscure transient, localized structural reorganizations during dynamic phase transitions. To overcome this limitation, we present a hierarchical dynamic evaluation framework driven by a novel topological and hypergraph reconstruction strategy. Instead of directly interpolating abstract network parameters, our method interpolates the underlying spatial geometry and rigorously recomputes the valid topological structures, ensuring physical fidelity. Along this geodesic, we introduce a set of multi-scale indicators: macroscopic metrics (Topological Distortion and Persistence Entropy) to capture global shifts, and a novel mesoscopic dual-perspective Hypergraph Entropy (node-perspective and edge-perspective) to detect highly sensitive, asynchronous local rewirings. We further propagate the cycle-level entropy change onto individual vertices to form a point-level topological field. Extensive evaluations on physical dynamical systems (Rayleigh-Van der Pol limit cycles, Double-Well cluster fusion), high-dimensional biological aggregation (D'Orsogna model), and longitudinal stroke fMRI data demonstrate the utility of combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis.

Yixin Wang, Ting Gao, Jinqiao Duan

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China · Center for Mathematical Science, Huazhong University of Science and Technology, Wuhan, China · Steklov-Wuhan Institute for Mathematical Exploration, Huazhong University of Science and Technology, China +2

Topological Data Analysis Optimal Transport Persistent Homology

Mar 15, 2026

Sampling Boltzmann distributions via normalizing flow approximation of transport maps

In a celebrated paper \cite{noe2019boltzmann}, Noé, Olsson, Köhler and Wu introduced an efficient method for sampling high-dimensional Boltzmann distributions arising in molecular dynamics via normalizing flow approximation of transport maps. Here, we place this approach on a firm mathematical foundation. We prove the existence of a normalizing flow between the reference measure and the true Boltzmann distribution up to an arbitrarily small error in the Wasserstein distance. This result covers general Boltzmann distributions from molecular dynamics, which have low regularity due to the presence of interatomic Coulomb and Lennard-Jones interactions. The proof is based on a rigorous construction of the Moser transport map for low-regularity endpoint densities and approximation theorems for neural networks in Sobolev spaces. Numerical simulations for a simple model system and for the alanine dipeptide molecule confirm that the true and generated distributions are close in the Wasserstein distance. Moreover we observe that the RealNVP architecture does not just successfully capture the equilibrium Boltzmann distribution but also the metastable dynamics.

Zia Ur Rehman, Gero Friesecke

Department of Mathematics, School of Computation, Information and Technology, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching, Germany

Boltzmann Sampling Normalizing Flows Molecular Dynamics Simulation Optimal Transport

Mar 13, 2026

Sobolev--Ricci Curvature

Ricci curvature is a fundamental concept in differential geometry for encoding local geometric structure, and its graph-based analogues have recently gained prominence as practical tools for reweighting, pruning, and reshaping network geometry. We propose Sobolev-Ricci Curvature (SRC), a graph Ricci curvature canonically induced by Sobolev transport geometry, which admits efficient evaluation via a tree-metric Sobolev structure on neighborhood measures. We establish two consistency behaviors that anchor SRC to classical transport curvature: (i) on trees endowed with the length measure, SRC recovers Ollivier-Ricci curvature (ORC) in the canonical W1 setting, and (ii) SRC vanishes in the Dirac limit, matching the flat case of measure-theoretic Ricci curvature. We demonstrate SRC as a reusable curvature primitive in two representative pipelines. We define Sobolev-Ricci Flow by replacing ORC with SRC in a Ricci-flow-style reweighting rule, and we use SRC for curvature-guided edge pruning aimed at preserving manifold structure. Overall, SRC provides a transport-based foundation for scalable curvature-driven graph transformation and manifold-oriented pruning.

Kyoichi Iwasaki, Tam Le, Hideitsu Hino

The Graduate University for Advanced Studies, SOKENDAI · The Institute of Statistical Mathematics

Optimal Transport

Mar 12, 2026

Sinkhorn-Drifting Generative Models

We establish a theoretical link between the recently proposed "drifting" generative dynamics and gradient flows induced by the Sinkhorn divergence. In a particle discretization, the drift field admits a cross-minus-self decomposition: an attractive term toward the target distribution and a repulsive/self-correction term toward the current model, both expressed via one-sided normalized Gibbs kernels. We show that Sinkhorn divergence yields an analogous cross-minus-self structure, but with each term defined by entropic optimal-transport couplings obtained through two-sided Sinkhorn scaling (i.e., enforcing both marginals). This provides a precise sense in which drifting acts as a surrogate for a Sinkhorn-divergence gradient flow, interpolating between one-sided normalization and full two-sided Sinkhorn scaling. Crucially, this connection resolves an identifiability gap in prior drifting formulations: leveraging the definiteness of the Sinkhorn divergence, we show that zero drift (equilibrium of the dynamics) implies that the model and target measures match. Experiments show that Sinkhorn drifting reduces sensitivity to kernel temperature and improves one-step generative quality, trading off additional training time for a more stable optimization, without altering the inference procedure used by drift methods. These theoretical gains translate to strong low-temperature improvements in practice: on FFHQ-ALAE at the lowest temperature setting we evaluate, Sinkhorn drifting reduces mean FID from 187.7 to 37.1 and mean latent EMD from 453.3 to 144.4, while on MNIST it preserves full class coverage across the temperature sweep. Project page: https://mint-vu.github.io/SinkhornDrifting/

Ping He, Om Khangaonkar, Hamed Pirsiavash +2

Department of Computer Science, Vanderbilt University · Department of Computer Science, University of California, Davis · Department of Computer Science, Purdue University

Sinkhorn Divergence Drifting Models Flow-Based Generative Modeling Optimal Transport One-Step Generative Modeling

Mar 12, 2026

Shape-of-You: Fused Gromov-Wasserstein Optimal Transport for Semantic Correspondence in-the-Wild

Semantic correspondence is essential for handling diverse in-the-wild images lacking explicit correspondence annotations. While recent 2D foundation models offer powerful features, adapting them for unsupervised learning via nearest-neighbor pseudo-labels has key limitations: it operates locally, ignoring structural relationships, and consequently its reliance on 2D appearance fails to resolve geometric ambiguities arising from symmetries or repetitive features. In this work, we address this by reformulating pseudo-label generation as a Fused Gromov-Wasserstein (FGW) problem, which jointly optimizes inter-feature similarity and intra-structural consistency. Our framework, Shape-of-You (SoY), leverages a 3D foundation model to define this intra-structure in the geometric space, resolving abovementioned ambiguity. However, since FGW is a computationally prohibitive quadratic problem, we approximate it through anchor-based linearization. The resulting probabilistic transport plan provides a structurally consistent but noisy supervisory signal. Thus, we introduce a soft-target loss dynamically blending guidance from this plan with network predictions to build a learning framework robust to this noise. SoY achieves state-of-the-art performance on SPair-71k and AP-10k datasets, establishing a new benchmark in semantic correspondence without explicit geometric annotations. Code is available at Shape-of-You.

Jiin Im, Sisung Liu, Je Hyeong Hong

Dept. Electronic Engineering, Hanyang University · Dept. Artificial Intelligence, Hanyang University

Semantic Correspondence Optimal Transport

Mar 12, 2026

MST-Direct: Matching via Sinkhorn Transport for Multivariate Geostatistical Simulation with Complex Non-Linear Dependencies

Multivariate geostatistical simulation requires the faithful reproduction of complex non-linear dependencies among geological variables, including bimodal distributions, step functions, and heteroscedastic relationships. Traditional methods such as the Gaussian Copula and LU Decomposition assume linear correlation structures and often fail to preserve these complex joint distribution patterns. We propose MST-Direct (Matching via Sinkhorn Transport), a novel algorithm based on Optimal Transport theory that uses the Sinkhorn algorithm to directly match multivariate distributions while preserving spatial correlation structures. The method processes all variables simultaneously as a single multidimensional vector, enabling relational matching across the full joint space rather than relying on pairwise linear dependencies.

Tchalies Bachmann Schmitz

Optimal Transport

Mar 11, 2026

COT-FM: Cluster-wise Optimal Transport Flow Matching

We introduce COT-FM, a general framework that reshapes the probability path in Flow Matching (FM) to achieve faster and more reliable generation. FM models often produce curved trajectories due to random or batchwise couplings, which increase discretization error and reduce sample quality. COT-FM fixes this by clustering target samples and assigning each cluster a dedicated source distribution obtained by reversing pretrained FM models. This divide-and-conquer strategy yields more accurate local transport and significantly straighter vector fields, all without changing the model architecture. As a plug-and-play approach, COT-FM consistently accelerates sampling and improves generation quality across 2D datasets, image generation benchmarks, and robotic manipulation tasks.

Chiensheng Chiang, Kuan-Hsun Tu, Jia-Wei Liao +2

National Taiwan University

Flow Matching Generative Modeling Flow-Based Generative Modeling Image Generation Optimal Transport

Mar 10, 2026

Unbalanced Optimal Transport Dictionary Learning for Unsupervised Hyperspectral Image Clustering

Hyperspectral images capture vast amounts of high-dimensional spectral information about a scene, making labeling an intensive task that is resistant to out-of-the-box statistical methods. Unsupervised learning of clusters allows for automated segmentation of the scene, enabling a more rapid understanding of the image. Partitioning the spectral information contained within the data via dictionary learning in Wasserstein space has proven an effective method for unsupervised clustering. However, this approach requires balancing the spectral profiles of the data, blurring the classes, and sacrificing robustness to outliers and noise. In this paper, we suggest improving this approach by utilizing unbalanced Wasserstein barycenters to learn a lower-dimensional representation of the underlying data. The deployment of spectral clustering on the learned representation results in an effective approach for the unsupervised learning of labels.

Joshua Lentz, Nicholas Karris, Alex Cloninger +1

Tufts University, Department of Mathematics, Medford, MA 02155, USA · University of California San Diego, Department of Mathematics, La Jolla, CA 92093, USA · University of California San Diego, Halıcıo˘glu Data Science Institute, La Jolla, CA 92093, USA

Image Clustering Dictionary Learning Hyperspectral Imaging Optimal Transport

Mar 9, 2026

Towards Effective and Efficient Graph Alignment without Supervision

Unsupervised graph alignment aims to find the node correspondence across different graphs without any anchor node pairs. Despite the recent efforts utilizing deep learning-based techniques, such as the embedding and optimal transport (OT)-based approaches, we observe their limitations in terms of model accuracy-efficiency tradeoff. By focusing on the exploitation of local and global graph information, we formalize them as the local representation, global alignment'' paradigm, and present a new global representation and alignment'' paradigm to resolve the mismatch between the two phases in the alignment process. We then propose Global representation and optimal transport-based Alignment (GlobAlign), and its variant, GlobAlign-E, for better Efficiency. Our methods are equipped with the global attention mechanism and a hierarchical cross-graph transport cost, able to capture long-range and implicit node dependencies beyond the local graph structure. Furthermore, GlobAlign-E successfully closes the time complexity gap between representative embedding and OT-based methods, reducing OT's cubic complexity to quadratic terms. Through extensive experiments, our methods demonstrate superior performance, with up to a 20% accuracy improvement over the best competitor. Meanwhile, GlobAlign-E achieves the best efficiency, with an order of magnitude speedup against existing OT-based methods.

Songyang Chen, Youfang Lin, Yu Liu +2

Beijing Jiaotong University, Beijing, 100044, P.R. China. · Peking University, Beijing, 100871, P.R. China.

Graph Matching Graph Representation Learning Optimal Transport

Mar 7, 2026

Conditional Unbalanced Optimal Transport Maps: An Outlier-Robust Framework for Conditional Generative Modeling

Conditional Optimal Transport (COT) problem aims to find a transport map between conditional source and target distributions while minimizing the transport cost. Recently, these transport maps have been utilized in conditional generative modeling tasks to establish efficient mappings between the distributions. However, classical COT inherits a fundamental limitation of optimal transport, i.e., sensitivity to outliers, which arises from the hard distribution matching constraints. This limitation becomes more pronounced in a conditional setting, where each conditional distribution is estimated from a limited subset of data. To address this, we introduce the Conditional Unbalanced Optimal Transport (CUOT) framework, which relaxes conditional distribution-matching constraints through Csiszár divergence penalties while strictly preserving the conditioning marginals. We establish a rigorous formulation of the CUOT problem and derive its dual and semi-dual formulations. Based on the semi-dual form, we propose Conditional Unbalanced Optimal Transport Maps (CUOTM), an outlier-robust conditional generative model built upon a triangular

c

-transform parameterization. We theoretically justify the validity of this parameterization by proving that the optimal triangular map satisfies the

c

-transform relationships. Our experiments on 2D synthetic and image-scale datasets demonstrate that CUOTM achieves superior outlier robustness and competitive distribution-matching performance compared to existing COT-based baselines, while maintaining high sampling efficiency.

Jiwoo Yoon, Kyumin Choi, Jaewoong Choi

Department of Statistics, Sungkyunkwan University · Department of Applied Artificial Intelligence, Sungkyunkwan University

Optimal Transport Conditional Generative Modeling

Mar 5, 2026

From Baselines to Transport Geodesics: Axiomatic Attribution via Optimal Generative Flows

Feature attributions often hide a critical modeling choice: they explain a prediction along a counterfactual path from a reference state to an input. Different baselines, interpolations, and generative trajectories define different paths and can therefor produce different explanations. We study this path ambiguity as a modeling problem. Our central question is whether the path can be chosen by the data-generating transport process, rather than by a hand-designed interpolation or by the sensitivity geometry of the model being explained. We separate attribution into fixed-path credit allocation and path selection. For a fixed path, we prove that the Aumann-Shapley line integral is the unique attribution rule under standard fixed-path axioms and explicit coordinate-trace regularity. For path selection, we minimize kinetic action over flows that transport a reference distribution to the data distribution, yielding a transport-geodesic attribution principle. We approximate this ideal with Rectified Flow and Reflow and derive stability bounds linking vector-field error to attribution error. Experiments show that lower-action, transport-consistent paths produce more stable and structured explanations, preserving competitive deletion faithfulness, without claiming data-manifold membership. Our code is available at https://github.com/cenweizhang/OTFlowSHAP.

Cenwei Zhang, Lin Zhu, Manxi Lin +1

Aalto University, Espoo, Finland · Alibaba, Hangzhou, China · Technical University of Denmark, Copenhagen, Denmark

Feature Attribution Shapley Value Attribution Optimal Transport

Mar 3, 2026

Transport Clustering: Solving Low-Rank Optimal Transport via Clustering

Optimal transport (OT) finds a least cost transport plan between two probability distributions using a cost matrix defined on pairs of points. Unlike standard OT, which infers unstructured pointwise mappings, low-rank optimal transport explicitly constrains the rank of the transport plan to infer latent structure. This improves statistical stability and robustness, yields sharper parametric rates for estimating Wasserstein distances adaptive to the intrinsic rank, and generalizes

K

-means to co-clustering. These advantages, however, come at the cost of a non-convex and NP-hard optimization problem. We introduce transport clustering, an algorithm to compute a low-rank OT plan that reduces low-rank OT to a clustering problem on correspondences obtained from a full-rank

\textit{transport registration}

step. We prove that this reduction yields polynomial-time, constant-factor approximation algorithms for low-rank OT: specifically, a

(1+γ)

approximation for negative-type metrics and a

(1+γ+\sqrt{2γ}\,)

approximation for kernel costs, where

γ\in [0,1]

denotes the approximation ratio of the optimal full-rank solution relative to the low-rank optimal. Empirically, transport clustering outperforms existing low-rank OT solvers on synthetic benchmarks and large-scale, high-dimensional datasets.

Henri Schmidt, Peter Halmos, Ben Raphael

Department of Computer Science, Princeton University, NJ, USA.

Optimal Transport Clustering

Mar 2, 2026

Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transport

Neural networks (NNs) often have critical behavioural trade-offs that are set at design time with hyperparameters-such as reward weights in reinforcement learning or quantile targets in regression. Post-deployment, however, user preferences can evolve, making initial settings undesirable, necessitating potentially expensive retraining. To circumvent this, we introduce the task of Hyperparameter Trajectory Inference (HTI): to learn, from observed data, how a NN's conditional output distribution changes with its hyperparameters, and construct a surrogate model that approximates the NN at unobserved hyperparameter settings. HTI requires extending existing trajectory inference approaches to incorporate conditions, exacerbating the challenge of ensuring inferred paths are feasible. We propose an approach based on conditional Lagrangian optimal transport, jointly learning the Lagrangian function governing hyperparameter-induced dynamics along with the associated optimal transport maps and geodesics between observed marginals, which form the surrogate model. We incorporate inductive biases based on the manifold hypothesis and least-action principles into the learned Lagrangian, improving surrogate model feasibility. We empirically demonstrate that our approach reconstructs NN outputs across various hyperparameter spectra better than other alternatives.

Harry Amad, Mihaela van der Schaar

Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK

Neural Surrogate Modeling Optimal Transport

Feb 28, 2026

COG: Confidence-aware Optimal Geometric Correspondence for Unsupervised Single-reference Novel Object Pose Estimation

Estimating the 6DoF pose of a novel object with a single reference view is challenging due to occlusions, view-point changes, and outliers. A core difficulty lies in finding robust cross-view correspondences, as existing methods often rely on discrete one-to-one matching that is non-differentiable and tends to collapse onto sparse key-points. We propose Confidence-aware Optimal Geometric Correspondence (COG), an unsupervised framework that formulates correspondence estimation as a confidence-aware optimal transport problem. COG produces balanced soft correspondences by predicting point-wise confidences and injecting them as optimal transport marginals, suppressing non-overlapping regions. Semantic priors from vision foundation models further regularize the correspondences, leading to stable pose estimation. This design integrates confidence into the correspondence finding and pose estimation pipeline, enabling unsupervised learning. Experiments show unsupervised COG achieves comparable performance to supervised methods, and supervised COG outperforms them.

Yuchen Che, Jingtu Wu, Hao Zheng +1

Institute of Science Tokyo · RIKEN · Tohoku University

Object Pose Estimation Unsupervised Learning Optimal Transport

Feb 27, 2026

Multimodal Optimal Transport for Training-free Temporal Segmentation in Surgical Robotics

Automated recognition of surgical phases and steps is a fundamental capability for intraoperative decision support, workflow automation, and skill assessment in robotic-assisted surgery. Existing approaches either depend on large-scale annotated surgical datasets or require expensive domain-specific pretraining on thousands of labeled videos, limiting their practical deployability across diverse robotic platforms and clinical environments. In this work, we propose TASOT (Text-Augmented Action Segmentation Optimal Transport), an annotation-free framework for surgical temporal segmentation that requires no task-specific annotations or surgical-domain pretraining. TASOT extends the Action Segmentation Optimal Transport (ASOT) formulation by incorporating temporally aligned textual descriptions generated directly from the input video, fusing visual and semantic cues within a unified unbalanced Gromov-Wasserstein optimal transport objective. Visual representations are extracted using DINOv3, while temporal captions produced by a vision-language model are encoded via CLIP and temporally aligned to individual frames, providing complementary semantic structure to the transport cost. We evaluate TASOT on three public surgical datasets and four benchmark settings spanning laparoscopic and robotic procedures, showing substantial improvements over the strongest zero-shot baselines: +18.9 F1 on Cholec80, +33.7 on AutoLaparo, +23.7 on StrasByPass70, and +4.5 on BernByPass70. These results suggest that fine-grained surgical workflow understanding in robotic settings can be achieved without manual training annotations or surgical-specific pretraining pipelines, offering a promising alternative for real-world robotic surgical systems.

Omar Mohamed, Edoardo Fazzari, Ayah Al-Naji +5

Authos are with the Dept. of Robotics, Mohamed bin Zayed University of AI, Abu Dhabi, UAE.

Temporal Action Segmentation Surgical Video Understanding Surgical Robotics Optimal Transport

Feb 25, 2026

Neural solver for Wasserstein Geodesics and optimal transport dynamics

In recent years, the machine learning community has increasingly embraced the optimal transport (OT) framework for modeling distributional relationships. In this work, we introduce a sample-based neural solver for computing the Wasserstein geodesic between a source and target distribution, along with the associated velocity field. Building on the dynamical formulation of the optimal transport (OT) problem, we recast the constrained optimization as a minimax problem, using deep neural networks to approximate the relevant functions. This approach not only provides the Wasserstein geodesic but also recovers the OT map, enabling direct sampling from the target distribution. By estimating the OT map, we obtain velocity estimates along particle trajectories, which in turn allow us to learn the full velocity field. The framework is flexible and readily extends to general cost functions, including the commonly used quadratic cost. We demonstrate the effectiveness of our method through experiments on both synthetic and real datasets.

Hailiang Liu, Yan-Han Chen

Department of Mathematics, Iowa State University, Ames, IA 50011, USA. · Department of Statistics, Iowa State University, Ames, IA 50011, USA.

Optimal Transport

Feb 25, 2026

Entropy-Controlled Flow Matching

Modern vision generators transport a base distribution to data through time-indexed measures, implemented as deterministic flows (ODEs) or stochastic diffusions (SDEs). Despite strong empirical performance, standard flow-matching objectives do not directly control the information geometry of the trajectory, allowing low-entropy bottlenecks that can transiently deplete semantic modes. We propose Entropy-Controlled Flow Matching (ECFM): a constrained variational principle over continuity-equation paths enforcing a global entropy-rate budget d/dt H(mu_t) >= -lambda. ECFM is a convex optimization in Wasserstein space with a KKT/Pontryagin system, and admits a stochastic-control representation equivalent to a Schrodinger bridge with an explicit entropy multiplier. In the pure transport regime, ECFM recovers entropic OT geodesics and Gamma-converges to classical OT as lambda -> 0. We further obtain certificate-style mode-coverage and density-floor guarantees with Lipschitz stability, and construct near-optimal collapse counterexamples for unconstrained flow matching.

Chika Maduabuchi

William & Mary, Williamsburg, VA, USA

Flow Matching Flow-Based Generative Modeling Entropy Regularization Optimal Transport

Feb 23, 2026

Mean-Field Path-Integral Diffusion: From Samples to Interacting Agents

Independent sample generation is the prevailing paradigm in modern diffusion-based generative models of AI. We ask a different question: can samples coordinate through shared population statistics to transport probability mass more efficiently? We introduce Mean-Field Path-Integral Diffusion (MF-PID), a framework in which samples are promoted to interacting agents whose drift depends self-consistently on the evolving population density. The coupling converts distribution matching into a McKean--Vlasov extension of the stochastic optimal transport problem, unifying generative modeling and multi-agent control under the same Hamilton--Jacobi--Bellman/Kolmogorov--Fokker--Planck duality. We identify two analytically tractable regimes: a Linear--Quadratic--Gaussian (LQG) benchmark in which the infinite-dimensional mean-field system reduces to a finite set of Riccati and linear ODEs, and a Gaussian-mixture regime governed by a piecewise-constant protocol that preserves closed-form solvability. For a quadratic interaction potential with schedule

β_t

and zero base drift we prove that the self-consistent MF guidance is the exact linear interpolant between initial and target global means -- a result that holds for arbitrary initial and target densities and any

β_t

. Applied to demand-response control of energy systems, where agents aggregated into an ensemble are energy consumers (e.g.\ thermal zones within a building), MF-PID achieves 19--24% reductions in cumulative control energy over independent-agent baselines while matching the prescribed terminal distribution exactly, and reveals how coordination redistributes actuation effort across heterogeneous sub-populations.

Michael Chertkov

Graduate Interdisciplinary Program in Applied Mathematics & Department of Mathematics, University of Arizona, Tucson, AZ, USA

Diffusion Models Multi-Agent Control Optimal Transport Mean Field Games