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.

May 21, 2026

Generative Modeling by Value-Driven Transport

We propose a new framework for generative modeling based on a discrete-time stochastic control formulation of measure transport. Adapting classic results from control theory, we formulate our problem as a linear program whose dual variables correspond to the optimal value function of the control problem, which directly encodes the optimal control policy. Exploiting this LP formulation, we develop an efficient simulation-free primal-dual algorithm for computing approximately optimal value functions and the associated value-driven transport (VDT) policies which approximate the true optimal policy. We show that well-trained VDT policies enjoy numerous favorable properties in comparison with other state-of-the-art methods based on flows, diffusions, or Schrödinger bridges: they lead to straight transport paths which can be simulated quickly and robustly, and can be enhanced in all the same ways as diffusion and flow-based models (e.g., conditional generation, classifier-free guidance, unpaired data-to-data translation are all easy to incorporate). We evaluate our methodology in a range of experiments, with results that indicate strong performance and good potential for scalability.

Pablo Moreno-Muñoz, Adrian Müller, Gergely Neu

Universitat Pompeu Fabra · Barcelona, Spain · ETH Zürich +2

Value Function Estimation Stochastic Optimal Control Generative Modeling Optimal Transport

May 20, 2026

On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures

Despite the remarkable empirical success of generative models, the available theory on their statistical accuracy in scientific computing remains largely pessimistic. This paper develops a theoretical framework for understanding the regularity of transport maps and the generalization properties of one-step Wasserstein-guided generative models for PDE-induced probability measures. We consider normalized target densities associated with linear elliptic and parabolic equations on bounded domains, as well as diffusion and Fokker--Planck equations on the torus. Under standard structural assumptions, we prove that these target measures satisfy doubling conditions. By combining this fact with regularity theory for optimal transport between doubling measures, we show that the optimal transport map from a uniform source measure to the target measure is Hölder continuous. This regularity yields an approximation-theoretic justification for one-step generative models that learn PDE-induced distributions via a single pushforward map. As a representative instance, we study DeepParticle and derive excess-risk bounds characterizing the discrepancy between the learned map and the population-optimal map. We also establish a robustness estimate under target shift and illustrate the theory with experiments which support the derived rates.

Likun Lin, Zhongjian Wang, Jack Xin +1

Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, P.R. China. · Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371. · Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA. +1

One-Step Generative Modeling Optimal Transport Generalization Bounds

May 20, 2026

UOTIP: Unbalanced Optimal Transport Map for Unpaired Inverse Problems

We investigate unpaired image inverse problems, a challenging setting where only independent, non-paired sets of noisy measurements and clean target signals are available for training. We propose a novel inverse problem solver based on Unbalanced Optimal Transport, called Unbalanced Optimal Transport Map for Inverse Problems (UOTIP). Our method formulates the reconstruction task, predicting clean target signals from noisy measurements, as learning a UOT Map from noisy measurement distribution to clean signal distribution by incorporating a likelihood-based cost function. By relaxing the exact marginal constraint, the UOT framework provides key advantages to our model: robustness to multi-level observation noise, adaptability to class imbalance between noisy and clean datasets, and generalizability to diverse noise-type scenarios. Furthermore, we theoretically demonstrate that incorporating a quadratic cost term ensures the existence and uniqueness of the transport map by satisfying the twist condition, even for ill-posed inverse problems. Our experiments demonstrate that UOTIP achieves state-of-the-art performance on unpaired image inverse problem benchmarks, across linear and nonlinear inverse problems.

Donggyu Lee, Taekyung Lee, Jaewoong Choi

Department of Mathematical Sciences, Seoul National University · IPAI (Interdisciplinary Program in Artificial Intelligence, Seoul National University) · Sungkyunkwan University

Image Reconstruction Optimal Transport Image Inverse Problems

May 20, 2026

Learning fMRI activations dictionaries across individual geometries via optimal transport

Dictionary learning is a powerful tool for creating interpretable representations. When applied to functional magnetic resonance imaging (fMRI) data, the resulting patterns of brain activity can be used for various downstream tasks, such as brain state classification or population-level analysis. However, a major challenge is the variability in brain geometry across individuals. This is usually addressed by projecting each individual brain geometry onto a common template, which removes subject-specific information. In this work, we introduce a novel approach to dictionary learning on fMRI data that explicitly accounts for this variability. We use the optimal transport-based Fused Gromov-Wasserstein (FGW) distance to compare graphs with different geometries and features. To address the challenge of computing multiple FGW distances for large graphs such as those arising from fMRI data, we rely on amortized optimization to learn a neural network that predicts an approximation of the optimal transport plans, which substantially reduces the computational cost. Additionally, we learn dictionary atoms that depend on the FGW trade-off parameter, which controls the balance between feature alignment and structural consistency. Numerical experiments on the HCP dataset demonstrate that the proposed approach captures different levels of geometric variability in the data and provides representations that preserve essential information.

Sonia Mazelet, Rémi Flamary, Bertrand Thirion

CMAP, Ecole Polytechnique · Palaiseau, France · Mind, Inria-Saclay

Learning to Optimize Geometric Representation Learning Dictionary Learning Neuroimaging Optimal Transport

May 19, 2026

Optimizing Computational-Statistical Runtime for Wasserstein Distance Estimation

Squared Wasserstein distance is a frequently used tool to measure discrepancy between probability distributions. This distance is typically computed between empirical measures of size

n

from two underlying random samples. Unfortunately, even in lower dimensional Euclidean space problems

\left( d \in \{2,3\} \right)

, algorithms for Wasserstein distance computation with approximate or exact precision guarantees scale poorly in the runtime as a function of

n

and the desired precision. In response, we consider the computational-statistical runtime, where the goal is to estimate from samples the Wasserstein distance between potentially smooth measures up to

ε

-additive error in expectation with respect to the sampling; we allow

O(1)

computational cost for collecting a sample. Towards this, we develop a Sample-Sketch-Solve paradigm where we introduce a regular cartesian grid sketch of the samples. We show that (especially under

α

-Hölder smooth distributions) this can compress the data without increasing asymptotic error, and also regularizes the structure which enables faster exact algorithms. Ultimately, we approximate

W_2^2(P,Q)

within

ε

error in

ε^{-\max(2,\frac{d+1+o(1)}{1+α})}

time for

0 < α< 1

Hölder smooth distributions

P,Q

on

(0,1)^{d}

; an optimal

Θ(ε^{-2})

for

α> 1/2

when

d=2

and nearly optimal as

α\to 1

when

d = 3

.

Peter Matthew Jacobs, Jeff M. Phillips

Department of Statistics University of Wisconsin-Madison Madison, WI 53706 · Kahlert School of Computing University of Utah Salt Lake City, UT 84101

Randomized Sketching Approximation Algorithms Optimal Transport Wasserstein Distance

May 19, 2026

Take It or Leave It: Intent-Controlled Partial Optimal Transport

While optimal transport (OT) enforces a rigid constraint by requiring two measures to be matched exactly, partial optimal transport relaxes this requirement by allowing mass to remain unmatched through a global budget, scalar rebate, or uniform rejection rule. However, many applications call for more structured, pointwise rejection mechanisms, where the decision to leave mass unmatched depends on side-specific reliability, support geometry, or external information about which components should participate in the comparison. We introduce intent-controlled partial optimal transport (IC-POT), a targeted generalization of partial transport that replaces the global rejection paradigm with pointwise rejection costs over both measures. We show that the resulting optimization problem admits a dual interpretation in terms of local acceptance thresholds and can be solved by recasting it as a balanced Kantorovich OT problem on an augmented support. Beyond theoretical analysis, we demonstrate the practical relevance of IC-POT in settings where rejection is driven by side information. In positive-unlabeled learning and open-partial domain adaptation, incorporating pointwise rejection rules that encode statistical structure improves fixed baseline pipelines. Finally, we motivate the use of IC-POT with a geophysical practical case: multi-modal satellite ocean measurements, for which physical and sensors priors naturally inform the rejection mechanism and define the retrieved comparable signal information.

Salil Parth Tripathi, Bertrand Chapron, Fabrice Collard +2

OceanDataLab · Ifremer · Université Bretagne Sud +1

Inverse Optimal Transport Optimal Transport

May 18, 2026

PACE: Geometry-Aware Bridge Transport for Single-Cell Trajectory Inference

Single-cell trajectory inference from destructive time-course snapshots is fundamentally ill-posed: neither cross-time cell correspondences nor continuous trajectories are observed, so the snapshot distributions alone do not uniquely determine the underlying dynamics. Existing optimal transport and flow-based methods typically couple cells by Euclidean proximity at observed clock times, which can misalign trajectories when development is asynchronous and cells sampled at the same experimental time occupy different latent pseudotime stages. We propose PACE, a trajectory inference framework that recovers geometry-consistent continuous transport dynamics from destructive time-course snapshots through three coupled components. First, PACE constructs a state- and time-dependent anisotropic Riemannian metric that assigns low transport cost along locally supported tangent directions while penalizing normal velocity components. Second, it alternates between refining cross-time couplings under the induced path-action cost and fitting endpoint-preserving neural bridges between adjacent snapshots. Third, it distills the learned bridge dynamics into a global continuous-time velocity field over cellular states. Across seven controlled and biological datasets covering nine held-out reconstruction experiments, PACE achieves the strongest overall reconstruction performance, reducing MMD, Wasserstein-1 distance, and Wasserstein-2 distance by 23.7% on average relative to the strongest competing baseline. PACE also improves RNA-velocity alignment by 15.4% on an embryoid body differentiation benchmark, without requiring explicit cell pairing, lineage tracing, or RNA-velocity supervision during training. Code is available at https://github.com/AI4Science-WestlakeU/PACE.

Chenglei Yu, Chuanrui Wang, Bangyan Liao +1

Zhejiang University · Department of Artificial Intelligence, School of Engineering, Westlake University

Trajectory Inference Single-Cell Transcriptomics Optimal Transport

May 18, 2026

Spherical Harmonic Optimal Transport: Application to Climate Models Comparisons

Optimal transport provides a powerful framework for comparing measures while respecting the geometry of their support, but comes with an expensive computational cost, hindering its potential application to real world use cases. On manifolds, convolutional algorithms based on the heat kernel have been proposed to alleviate this cost, but their theoretical properties remain largely unexplored. We establish that the heat kernel cost converges to the optimal transport cost as time vanishes in the balanced and unbalanced cases. In the specific case of the 2-sphere

\mathbb{S}^2

, we ensure that the associated Sinkhorn divergences retains the desirable geometric and analytic properties of classical optimal transport discrepancies. Moreover, we leverage the harmonic structure of the sphere to derive a fast Sinkhorn algorithm, requiring only

\mathcal{O}(n)

memory and

\mathcal{O}(n^{3/2})

time per iteration, with fully dense GPU-friendly operations. We validate its computational efficiency on synthetic data, and discuss its potential use in the evaluation of global climate models, providing both spatial and seasonal insights into models performances.

Pierre Houédry, Iskander Legheraba, Léo Buecher +1

INRIA Rennes · LPHI, UMR 5294, CNRS, INSERM · IRISA, UMR 6074, CNRS

Sinkhorn Divergence Kernel Methods Optimal Transport Wasserstein Distance

May 17, 2026

A Distribution Matching Approach to Neural Piano Transcription with Optimal Transport

This paper describes a novel paradigm that formalizes automatic piano transcription (APT) as an optimal transport (OT) problem, not as a frame-level multi-label binary classification problem. Our method learns to minimize the cost of transporting a predicted distribution of note events to the ground-truth distribution over time and frequency. The OT loss can thus accommodate temporal misalignment, leading to perceptually relevant optimization. We also propose a convolutional recurrent neural network (CRNN) with a harmonics-aware attention mechanism to capture the spectro-temporal dependencies inherent in music.Our experiments using the MAESTRO dataset showed that our method attained a state-of-the-art performance in onset detection. We confirmed the versatility of the OT loss in application to existing models.

Weixing Wei, Raynaldi Lalang, Dichucheng Li +1

Graduate School of Informatics, Kyoto University, Japan · Graduate School of Engineering, Kyoto University, Japan · Independent Researcher, Hong Kong, China

Convolutional RNNs Automatic Music Transcription Music Information Retrieval Optimal Transport

May 17, 2026

Scalable Bi-causal Optimal Transport via KL Relaxation and Policy Gradients

Bi-causal optimal transport (OT) is a natural framework for comparing and coupling stochastic processes under nonanticipative information constraints, with important applications in robust finance, sequential uncertainty quantification, and multistage stochastic optimization. In particular, a learned bi-causal coupling naturally serves as a simulator for generating joint sample paths that respect both prescribed marginal laws and the underlying information flow. Its practical use, however, is limited by the computational difficulty of enforcing bi-causal coupling constraints over path space, especially for continuous distributions and long horizons. We develop a scalable stochastic-optimization framework for computing bi-causal OT couplings under general marginals. Our approach introduces a Kullback--Leibler (KL)-penalized relaxation that replaces hard marginal constraints with tractable divergence penalties while preserving the recursive structure of the problem. We establish dynamic programming principles for both the original and relaxed formulations, prove that the relaxed problem converges to the original bi-causal OT problem as the penalty grows, and derive explicit policy-gradient representations for the relaxed objective. Building on these results, we propose a practical policy-gradient algorithm with unbiased mini-batch estimators, variance reduction, and nonasymptotic regret guarantees. Numerical experiments show that the method accurately captures marginal laws and temporal dependence, and performs well in applications including robust subhedging and time series statistical downscaling. These results provide a scalable computational approach to bi-causal OT and broaden its applicability in settings where nonanticipative information constraints are essential.

Haoyang Cao, Jesse Hoekstra, Renyuan Xu +2

Department of Applied Mathematics and Statistics, Data Science and AI Institute, and Mathematical Institute for Data Science, Johns Hopkins University. · Department of Statistics, Oxford-Man Institute of Quantitative Finance, and Nuffield College, University of Oxford. · Management Science & Engineering Department, Stanford University. +2

Stochastic Optimization Optimal Transport

May 15, 2026

Multiscale Supervised Unbalanced Optimal Transport Flow Matching

Unbalanced optimal transport (UOT) provides a principled framework for modeling single-cell transitions and birth-death dynamics, but its high computational cost limits scalability to large-scale datasets. Although single-cell data often contain hierarchical annotations and known transition priors, existing UOT approximations rarely exploit this multiscale structure or prior knowledge. We introduce Multiscale Supervised Unbalanced Optimal Transport Flow Matching (MUST-FM), a simulation-free framework that scales UOT by leveraging hierarchical data structure. MUST-FM further supports an optional supervised formulation that incorporates transition priors, such as cell lineages, to guide the learning of displacement fields and mass variations. Experiments show that MUST-FM reduces computational overhead while achieving robust and biologically meaningful trajectory inference, enabling dynamic modeling of atlas-scale single-cell datasets.

Qiangwei Peng, Lezhi Chen, Peijie Zhou

College of Electrical Engineering Sichuan University, Sichuan, China · AI for Science Institute, Beijing, China

Flow Matching Single-Cell Transcriptomics Optimal Transport

May 14, 2026

Distance-Matrix Wasserstein Statistics for Scalable Gromov--Wasserstein Learning

Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport problem and is difficult to estimate at scale. We propose Distance-Matrix Wasserstein (DMW), a hierarchy of Wasserstein statistics comparing laws of random finite distance matrices. Rather than optimizing a global point-level alignment, DMW samples

n

points from each space, records their pairwise distances, and transports the resulting matrix laws. We prove that DMW is a relaxation and lower bound of GW, and establish a reverse approximation inequality: the GW--DMW gap is controlled by the Wasserstein error of approximating each original measure with

n

samples. Hence population DMW converges to GW as sampled subspaces become dense. We further give finite-sample bounds, including intrinsic-dimensional rates that depend on the data manifold rather than the ambient matrix dimension

\binom n2

. For scalable computation, we introduce sliced and multi-scale DMW; for

p=1

, the sliced multi-scale dissimilarity yields positive-definite exponential kernels. Experiments on synthetic metric spaces, scalability benchmarks, graph classification, and two-sample testing validate the theory and demonstrate an interpretable GW-style proxy for structural comparison.

Ao Xu, Tieru Wu

School of Artificial Intelligence Jilin University No. 2699, Qianjin Street, Chaoyang District Changchun 130012, China · Zhongguancun Academy Daniufang 2nd Ring Road, Haidian District2026 Beijing 100094, China · School of Artificial IntelligenceMay Jilin University No. 2699, Qianjin Street, Chaoyang District14 Changchun 130012, China

Optimal Transport Wasserstein Distance

May 13, 2026

Min Generalized Sliced Gromov Wasserstein: A Scalable Path to Gromov Wasserstein

We propose min Generalized Sliced Gromov--Wasserstein (min-GSGW), a sliced formulation for the Gromov--Wasserstein (GW) problem using expressive generalized slicers. The key idea is to learn coupled nonlinear slicers that assign compatible push-forward values to both input measures, so that monotone coupling in the projected domain lifts to a transport plan evaluated against the GW objective in the original spaces. The resulting plan induces a GW objective value, and min-GSGW minimizes this cost directly in the original spaces. We further show that min-GSGW is rigid-motion invariant, a crucial property for geometric matching and shape analysis tasks. Our contributions are threefold: 1) we introduce generalized slicers into the sliced GW framework, 2) we construct a slicing-based efficient GW transport plan; and 3) we develop an amortized variant that replaces per-instance optimization with a learned slicer for unseen input pairs. We perform experiments on animal mesh matching, horse mesh interpolation, and ShapeNet part transfer. Results show that min-GSGW produces meaningful geometric correspondences and GW objective values at substantially lower computational cost than existing GW solvers.

Ashkan Shahbazi, Xinran Liu, Ping He +1

Department of Computer Science, College of Connected Computing, Vanderbilt University · Department of Electrical and Computer Engineering, Vanderbilt University

Non-Rigid Shape Correspondence Optimal Transport Wasserstein Distance

May 13, 2026

Path-independent Flow Matching for Multi-parameter Generative Dynamics

Flow Matching is a powerful framework for learning transport maps between probability distributions. Yet its standard single-parameter formulation is not designed to capture multi-parameter variations where the resulting transport should be path-independent. Path independence is crucial because it ensures that transformations depend only on the initial and target distributions, not on the specific path. In this work, we introduce Path-independent Flow Matching (PiFM), a method for learning vector fields whose induced flows yield path-independent transport between distributions. We show that PiFM generalizes Flow Matching to higher-dimensional parameter domains while enforcing structural conditions that ensure consistency of composed transformations. In addition, we show that, under suitable assumptions, PiFM approximates the Wasserstein barycenter, linking the framework to a notion of distributional interpolation. To enable practical training, we propose a tractable, simulation-free objective that regresses onto multi-parameter conditional probability paths. We showcase empirically that PiFM outperforms other approaches on both synthetic and real world data in interpolating path-independent trajectories and generating desired out of distribution samples.

Francisco Téllez, AmirHossein Zamani, Philippe Martin +5

Université de Montréal, Mila · Concordia University, Mila · Concordia University +2

Flow Matching Flow Map Learning Generative Modeling Flow-Based Generative Modeling Optimal Transport

May 13, 2026

Coupling-Informed Transport Maps for Bayesian Filtering in Nonlinear Dynamical Systems

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

Dengfei Zeng, Lijian Jiang, Shuyu Sun +1

School of Mathematical Sciences, Tongji University, Shanghai, 200092 China · School of Mathematical Sciences, Tongji University, Shanghai, 200092 China, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, 200092 China

Dynamical Systems Simulation-Based Inference Bayesian Filtering Optimal Transport

May 13, 2026

Pareto-Guided Optimal Transport for Multi-Reward Alignment

Text-to-image generation models have achieved remarkable progress in preference optimization, yet achieving robust alignment across diverse reward models remains a significant challenge. Existing multi-reward fusion approaches rely on weighted summation, which is costly to tune and insufficient for balancing conflicting objectives. More critically, optimization with reward models is highly susceptible to reward hacking, where reward scores increase while the perceived quality of generated images deteriorates. We demonstrate that optimizing against a unified global target under heterogeneous reward upper bounds can induce reward hacking, a risk further exacerbated by the inherent instability of weak reward models. To mitigate this, we propose a Pareto Frontier-Guided Optimal Transport (PG-OT) framework. Our method constructs a prompt-specific Pareto frontier and maps dominated samples toward it via distribution-aware optimal transport. Furthermore, we develop both online and offline optimization strategies tailored to diverse reward signal characteristics. To provide a more rigorous assessment, we introduce the Joint Domination Rate (JDR) and Joint Collapse Rate (JCR) as principled metrics to quantify multi-reward synergy and reward hacking. Experimental results show that our approach outperforms strong baselines with an 11% gain in JDR and achieves a near 80% win rate in human evaluations.

Ying Ba, Tianyu Zhang, Mohan Zhou +5

Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China · Beijing Key Laboratory of Research on Large Models and Intelligent Governance · Engineering Research Center of Next-Generation Intelligent Search and Recommendation, MOE 4Tomorrow Advancing Life +2

Diffusion Model Alignment Reward Hacking Multi-Objective Language Model Alignment T2I Diffusion Models Optimal Transport

May 12, 2026

Expected Batch Optimal Transport Plans and Consequences for Flow Matching

Solving optimal transport (OT) on random minibatches is a common surrogate for exact OT in large-scale learning. In flow matching (FM), this surrogate is used to obtain OT-like couplings that can straighten probability paths and reduce numerical integration cost. Yet, the population-level coupling induced by repeated minibatch OT remains only partially understood. We formalize this coupling as the expected batch OT plan

\overlineπ_{k}

, obtained by averaging empirical OT plans over independent minibatches of size

k

. We then establish its large-batch consistency and, in the semidiscrete case relevant to generative modeling, derive rates for both the transport-cost bias and the convergence of

\overlineπ_{k}

to the OT plan. For FM, this yields a population coupling whose induced velocity field is regular enough to define a unique flow from the source to the discrete target. We finally quantify how OT batch size interacts with numerical integration in a tractable two-atom model and in synthetic and image experiments.

Samuel Boïté, Julie Delon, Kimia Nadjahi

1DMA, ENS Paris · Université Paris Cité · DI, ENS Paris, CNRS

Flow Matching Optimal Transport

May 12, 2026

Fused Gromov-Wasserstein Distance with Feature Selection

Fused Gromov-Wasserstein (FGW) distances provide a principled framework for comparing objects by jointly aligning structure and node features. However, existing FGW formulations treat all features uniformly, which limits interpretability and robustness in high-dimensional settings where many features may be irrelevant or noisy. We introduce FGW distances with feature selection, which incorporate adaptive feature suppression weights into the FGW objective to selectively downweight or suppress differentiating features during alignment. We propose two approaches: (1) regularized FGW with Lasso and Ridge penalties, and (2) FGW with simplex-constrained weights, including groupwise extensions. We analyze the resulting models and establish their key theoretical properties, including bounds relative to classical FGW and Gromov-Wasserstein distances, and metric behavior. An efficient alternating minimization algorithm is developed. Experiments illustrate how feature suppression enhances interpretability and reveals task-relevant structure, with a special application to computational redistricting.

Harlin Lee, Ying Yu, Mingxin Li +1

School of Data Science and Society · University of North Carolina at Chapel Hill · Division of Computational Social Science +5

Feature Selection Optimal Transport Wasserstein Distance

May 12, 2026

QDSB: Quantized Diffusion Schrödinger Bridges

Learning generative models in settings where the source and target distributions are only specified through unpaired samples is gaining in importance. Here, one frequently-used model are Schrödinger bridges (SB), which represent the most likely evolution between both endpoint distributions. To accelerate training, simulation-free SBs avoid the path simulation of the original SB models. However, learning simulation-free SBs requires paired data; a coupling of the source and target samples is obtained as the solution of the entropic optimal transport (OT) problem. As obtaining the optimal global coupling is infeasible in many practical cases, the entropic OT problem is iteratively solved on minibatches instead. Still, the repeated cost remains substantial and the locality can distort the global transport geometry. We propose quantized diffusion Schrödinger bridges (QDSB), which compute the endpoint coupling on anchor-quantized endpoint distributions and lift the resulting plan back to original data points through cell-wise sampling. We show that the regularized optimal coupling is stable w.r.t. anchor quantization, with an error controlled by the quality of the anchor approximation. In real-world experiments, QDSB matches the sample quality of existing baselines, requiring substantially less time. Code and data are available at github.com/mathefuchs/qdsb.

Tobias Fuchs, Florian Kalinke, Nadja Klein

Karlsruhe Institute of Technology, Karlsruhe, Germany

Schrödinger Bridges Diffusion Model Acceleration Optimal Transport

May 12, 2026

OTT-Vid: Optimal Transport Temporal Token Compression for Video Large Language Models

As Video Large Language Models (Video-LLMs) scale to longer and more complex videos, their inference cost grows rapidly due to the large volume of visual tokens accumulated across frames. Training-free token compression has emerged as a practical solution to this bottleneck. However, existing temporal compression methods rely primarily on cross-frame token similarity or segmentation heuristics, overlooking each token's semantic role within its frame and failing to adapt compression strength to the compressibility of each frame pair. In this work, we propose OTT-Vid, a transport-derived allocation framework for temporal token compression. Our approach consists of two stages: spatial pruning identifies representative content within each frame, and optimal transport (OT) is then solved between neighboring frames to estimate temporal compressibility. We formulate this OT with non-uniform token mass, which protects semantically important tokens from aggressive compression, and a locality-aware cost that captures both feature and spatial disparities. The resulting transport plan jointly balances token importance and matching cost, while its total cost defines the transport difficulty of each frame pair, which we use to allocate compression budgets dynamically. Experiments on six benchmarks spanning video question answering and temporal grounding show that OTT-Vid preserves 95.8% of VQA and 73.9% of VTG performance while retaining only 10% of tokens, consistently outperforming existing state-of-the-art training-free compression methods.

Minseok Kang, Minhyeok Lee, Jungho Lee +6

Yonsei University · LG Electronics · KIST

Efficient VLM Inference Video-Language Models Video Token Compression Optimal Transport Visual Token Compression

May 11, 2026

Optimal and Scalable MAPF via Multi-Marginal Optimal Transport and Schrödinger Bridges

We consider anonymous multi-agent path finding (MAPF) where a set of robots is tasked to travel to a set of targets on a finite, connected graph. We show that MAPF can be cast as a special class of multi-marginal optimal transport (MMOT) problems with an underlying Markovian structure, under which the exponentially large MMOT collapses to a linear program (LP) polynomial in size. Focusing on the anonymous setting, we establish conditions under which the corresponding LP is feasible, totally unimodular, and consequently, yields min-cost, integral

(\{0,1\})

transports that do not overlap in both space and time. To adapt the approach to large-scale problems, we cast the MAPF-MMOT in a probabilistic framework via Schrödinger bridges. Under standard assumptions, we show that the Schrödinger bridge formulation reduces to an entropic regularization of the corresponding MMOT that admits an iterative Sinkhorn-type solution. The Schrödinger bridge, being a probabilistic framework, provides a shadow (fractional) transport that we use as a template to solve a reduced LP and demonstrate that it results in near-optimal, integral transports at a significant reduction in complexity. Extensive experiments highlight the optimality and scalability of the proposed approaches.

Usman A. Khan, Joseph W. Durham

Amazon Robotics, Boston, MA, USA · Boston College, Chestnut Hill, MA, USA · UAK holds concurrent appointments as an Amazon Scholar with Amazon Robotics and as a Professor of Computer Science at Boston College

Multi-Agent System Optimization Multi-Agent Path Finding Optimal Transport

May 11, 2026

Implicit Neural Optimal Transport via Fixed-Point Optimization

We propose an implicit neural formulation of optimal transport that eliminates adversarial min--max optimization and multi-network architectures commonly used in existing approaches. Our key idea is to parameterize a single potential in the Kantorovich dual and reformulate the associated c-transform as a proximal fixed-point problem. This yields a stable single-network framework in which dual feasibility is enforced exactly through proximal optimality conditions rather than adversarial training. Despite the inner fixed-point computation, gradients can be computed without differentiating through the fixed-point iterations, enabling efficient training without requiring implicit differentiation. We further establish convergence of stochastic gradient descent. The resulting framework is efficient, scalable, and broadly applicable: it simultaneously recovers forward and backward transport maps and naturally extends to class-conditional settings. Experiments on high-dimensional Gaussian benchmarks, physical datasets, and image translation tasks demonstrate strong transport accuracy together with improved training stability and favorable computational and memory efficiency.

Yesom Park, Eric Gelphman, Stanley Osher +1

Department of Mathematics, University of California, Los Angeles · Department of Applied Mathematics and Statistics, Colorado School of Mines

Optimal Transport

May 11, 2026

The Observable Wasserstein Distance

We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets. Analogous to the sliced Wasserstein distance in

\mathbb{R}^d

, our approach projects measures onto the real line via 1-Lipschitz observables and computes the Wasserstein distances between the resulting pushforward distributions. We define a hierarchy of pseudo-metrics by restricting observables to a nested chain of subspaces. A central theoretical contribution is an injectivity result linking the metric covering dimension of the support of a measure to the specific order in the hierarchy that guarantees unique recovery. This serves as a metric-space analogue to the Cramér-Wold Device for Euclidean distributions. We demonstrate that this hierarchy offers a tunable trade-off between sharpness as a lower bound on the Wasserstein distance and computational efficiency. We also present a discrete computational model for finite grids and numerical experiments validating the efficacy and utility of these approximations.

Edivaldo Lopes dos Santos, Leandro Vicente Mauri, Washington Mio +1

1Departamento de Matem´atica, Universidade Federal de S˜ao Carlos, Brazil · Department of Mathematics, Florida State University, USA

Sliced Wasserstein Distance Optimal Transport Wasserstein Distance

May 9, 2026

Geometrically Constrained Stenosis Editing in Coronary Angiography via Entropic Optimal Transport

The scarcity of high-quality imaging data for coronary angiography (CAG) stenosis limits the clinical translation of automated stenosis detection. Synthetic stenosis data provides a practical avenue to augment training sets, improving data quality, diversity, and distributional coverage, and enhancing detection precision and generalization. However, diffusion-based editing commonly relies on soft guidance in a noise-initialized reverse process, offering limited pixel-level precision and structure preservation. We propose the OT-Bridge Editor, which reframes localized editing as a constrained entropic optimal transport (OT) problem and leverages geometric information to steer the generation path, enabling stronger geometric control. Extensive experiments show that our synthesized angiograms consistently improve downstream stenosis detection, yielding substantial relative gains of 27.8% on the public ARCADE benchmark and 23.0% on our multi-center dataset, supported by consistent qualitative results.

Jialin Li, Zhuo Zhang, Yue Cao +4

School of Electrical and Information Engineering, Tianjin University, Tianjin, China.

Synthetic Data Augmentation Medical Imaging Coronary Angiography Medical Image Synthesis Image Editing Optimal Transport

May 9, 2026

cuRegOT: A GPU-Accelerated Solver for Entropic-Regularized Optimal Transport

Optimal transport (OT) has emerged as a fundamental tool in modern machine learning, yet its computational cost remains a significant bottleneck for large-scale applications. While harnessing the massive parallelism of modern GPU hardware is critical for efficiency, the de facto standard Sinkhorn algorithm, despite its ease of parallelization, often suffers from slow convergence in challenging problems. More recently, the sparse-plus-low-rank quasi-Newton method offers a balance between convergence rate and per-iteration complexity; however, its efficiency on GPUs is severely hindered by the serial nature of sparse matrix symbolic analysis and irregular memory access patterns. To bridge this gap, we present cuRegOT, a high-performance GPU solver tailored for entropic-regularized OT. We introduce a suite of algorithmic and architectural optimizations, including an amortized symbolic analysis strategy to mitigate CPU bottlenecks, an asynchronous Sinkhorn iterates generation mechanism, and a fused kernel for bandwidth-efficient gradient evaluation. These strategies are backed by rigorous theoretical guarantees ensuring algorithmic convergence. Extensive numerical experiments demonstrate that cuRegOT achieves significant speedups over state-of-the-art GPU-based solvers across a variety of benchmark tasks.

Yixuan Qiu

School of Statistics and Data Science & Institute of Big Data Research Shanghai University of Finance and Economics

GPU Kernel Optimization Quasi-Newton Methods Optimal Transport

May 8, 2026

Sliced Inner Product Gromov-Wasserstein Distances

The Gromov-Wasserstein (GW) problem provides a framework for aligning heterogeneous datasets by matching their intrinsic geometry, but its statistical and computational scaling remains an issue for high-dimensional problems. Slicing techniques offer an appealing route to scalability, but, unlike Wasserstein distances, GW problems do not generally admit closed-form solutions in one-dimension. We resolve this problem for the GW problem with inner product cost (IGW), propose a sliced IGW distance that enjoys a natural rotational invariance property, and comprehensively study its structural and computational properties. Numerical experiments validating our theory are presented, followed by applications to heterogeneous clustering of text data and language model representation comparison.

Xiaoyun Gong, Gabriel Rioux, Ziv Goldfeld

Optimal Transport Wasserstein Distance

May 8, 2026

Quantile-Coupled Flow Matching for Distributional Reinforcement Learning

Unlike standard expected-return Reinforcement Learning (RL), Distributional RL (DRL) models the full return distribution, making it better-suited for uncertainty-aware and risk-sensitive decision-making. Conditional Flow Matching (CFM) critics have recently attracted attention for modelling continuous, multi-modal return distributions. Despite this interest, there remains a substantial metric mismatch: DRL theory relies on the distributional Bellman operator being contractive in the

p

-Wasserstein distance, yet existing CFM critics are trained with arbitrary source-target couplings, so their flow-matching losses are not Wasserstein-aligned surrogates for matching Bellman target return distributions. In this work, we address this mismatch by proposing FlowIQN, a CFM critic that sorts source and Bellman target samples within each mini-batch to approximate the monotone optimal transport coupling, replacing arbitrary pairings with quantile-aligned flow paths. We prove that the loss of our quantile-coupled CFM critic yields a Wasserstein-aligned approximate projection compatible with the foundations of DRL. To our knowledge, FlowIQN is the first flow-matching distributional critic with an explicit Wasserstein-aligned projection guarantee. We further extend FlowIQN with shortcut models for efficient inference. Empirical results show that FlowIQN improves Wasserstein return-distribution accuracy over other CFM critics. It also yields competitive performance on offline RL benchmarks across multiple policy extraction methods, providing a theoretically grounded CFM critic that is readily compatible with DRL pipelines. Code: https://github.com/ori-goals/flowIQN.

Michael Groom, Victor-Alexandru Darvariu, Lars Kunze +2

Oxford Robotics Institute, University of Oxford, Oxford, United Kingdom · Bristol Robotics Laboratory, University of the West of England, Bristol, United Kingdom · Dyson Institute of Engineering & Technology, Malmesbury, United Kingdom

Distributional RL Conditional Flow Matching Optimal Transport

May 8, 2026

Sinkhorn Treatment Effects: A Causal Optimal Transport Measure

We introduce the Sinkhorn treatment effect, an entropic optimal transport measure of divergence between counterfactual distributions. Unlike classical quantities such as the average treatment effect, this measure captures differences across entire distributions. We analyze this divergence as a statistical functional and show it can be written as a smooth transformation of counterfactual mean embeddings with an appropriate kernel. This characterization allows us to establish first-order pathwise differentiability in general, and second-order pathwise differentiability under the null hypothesis of equal counterfactual distributions. Leveraging this smoothness, we construct debiased estimators and use them to obtain asymptotically valid tests for distributional treatment effects with a fixed entropic regularization parameter. Because the power of the test depends on this unknown parameter, we further propose an aggregated test that combines evidence across a grid of regularization choices. Experiments on simulated and image data demonstrate the practical advantages of our estimator and testing procedure.

Medha Agarwal, Alex Luedtke

Department of Statistics, University of Washington, Seattle, WA, USA · Department of Health Care Policy, Harvard Medical School, Boston, MA, USA

Causal Effect Estimation Sinkhorn Divergence Causal Inference Optimal Transport Statistical Hypothesis Testing Wasserstein Distance

May 8, 2026

Generalized Wasserstein Flow Matching: Nested Couplings for Point Cloud Generation

We formulate flow matching on probability measures over probability measures. Directly from the Wasserstein-on-Wasserstein (WoW) continuity equation, we show that admissible nested outer and inner couplings induce a valid metameasure flow and non-local velocity field. Validity requires neither optimal plans nor a smooth manifold. Practically, this opens up the design space of multisample flow matching: exact Wasserstein plans are distinguished by energy and straightness, while sliced and lazy linear plans yield scalable alternatives. Point-cloud experiments, including controlled Bures-Wasserstein and Euclidean minibatch baselines, demonstrate the resulting geometry-cost trade-off.

Moritz Piening, Richard Duong, Gabriele Steidl

Institut für Mathematik, Technische Universität Berlin, Germany

Flow Matching Generative Modeling Point Cloud Generation Flow-Based Generative Modeling Optimal Transport

May 7, 2026

Distributional Process Reward Models: Calibrated Prediction of Future Rewards via Conditional Optimal Transport

Inference-time scaling methods rely on Process Reward Models (PRMs), which are often poorly calibrated and overestimate success probabilities. We propose, to our knowledge, the first use of conditional optimal transport for calibrating PRMs, modifying conditional OT (CondOT) map learning \cite{bunne2022supervised} to estimate a monotonic conditional quantile function over success probabilities estimated by the PRM, conditioned on PRM hidden states. This yields structurally valid quantile estimates and enables efficient extraction of confidence bounds at arbitrary levels, which we integrate into the instance-adaptive scaling (IAS) framework of \cite{park2025know}. We evaluate on mathematical reasoning benchmarks spanning moderate-difficulty problems (MATH-500) and harder out-of-distribution problems (AIME). For PRMs with reliable ranking signals, our method substantially improves calibration over both uncalibrated PRMs and quantile regression. On downstream Best-of-N IAS performance, our method generally improves over uncalibrated PRMs. These results establish conditional optimal transport as another principled and practical approach to PRM calibration, offering structural guarantees and flexible uncertainty estimation.

Rachel Ma, Dylan Hadfield-Menell, Kristjan Greenewald

MIT CSAIL · MIT-IBM Watson AI Lab · IBM Research

Process Reward Models Inference-Time Scaling Uncertainty Calibration Optimal Transport