Optimal Transport

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Overview

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

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291 papers50 papers in the last four weeks, up 22% on the four weeks before. 0.3% of all new papers.

Aug 4, 2026

One Knob to Rule Them All: A Unified Optimal Transport View of Cold-Start Active Learning

Cold-Start Active Learning (CSAL) aims to select a valuable subset from an unlabeled pool without any prior knowledge or human assistance. Existing methods take diverse routes based on typicality, coverage, or diversity. Each rests on its own inductive bias and therefore performs well on some tasks yet poorly on others. We argue that the real challenge is not to design yet another selection heuristic, but to make CSAL adapt automatically to the data and task at hand. To this end, we revisit CSAL through the lens of optimal transport. First, we propose a generalized transport selection framework that reveals the shared allocation structure of existing methods and exactly subsumes representative formulations. Second, we introduce a theoretical analysis that characterizes the trade-off controlled by entropic regularization and establishes a task-agnostic minimax bound for cold-start selection. These results provide a principled foundation for adapting the regularization strength to the unlabeled data. Third, we derive a data-adaptive regularization rule and present a novel Sinkhorn-based CSAL algorithm, termed εε-Adaptive Selection (εε-AS). Extensive experiments on six public datasets and multiple annotation budgets show that εε-AS consistently achieves state-of-the-art performance. On ImageNet-1k, it improves the average accuracy over ActiveFT by 1.29% while reducing selection time by 56.2%. Code will be released at https://github.com/Z-yiwei/OT-CSAL

Ning Zhu, Xiaochuan Ma, Juntao Xu +4

Glasgow College, University of Electronic Science and Technology of China · School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China · School of Mathematical Sciences, University of Electronic Science and Technology of China

Active LearningTraining Data SelectionOptimal Transport
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Aug 3, 2026

Computational and Statistical Guarantees of the c-Rectified flow

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying c-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. More generally, for a Gaussian to log-concave coupling, the ordinary rectified flow can reach a fixed point after one step from independent initialization. In contrast, under suitable regularity and uniform-integrability assumptions, iterative c-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative c-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension d≥3d \ge 3 and a nearly parametric rate for d=1,2d=1,2.

Leda Wang, Zhehao Xu, Qiang Liu +1

Department of Statistics and Data Science, Yale University · Department of Computer Science, University of Texas at Austin

Rectified FlowOptimal Transport
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Aug 3, 2026

Beckmann Transport Models: From Autonomous Flows to One-Step Maps

We propose an instantiation of flow matching that relies on a time-independent velocity field (an autonomous flow) to exactly map between two distributions, so long as the target is singular, i.e.\ supported on a lower-dimensional data manifold. We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation, which can be used to learn the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem. Their construction provides a unifying framework that recovers, for instance, the closed-form Poisson-flow generative model and equilibrium matching with a quadratic flow-matching regression loss. We illustrate how this theory corrects inconsistencies in existing methods and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet 256x256.

Lee Cheuk-Kit, Florentin Coeurdoux, Yuyuan Chen +5

1Harvard University · 2Capital Fund Management · University of Pennsylvania +2

Flow MatchingOne-Step Generative ModelingFlow-Based Generative ModelingOptimal Transport
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Jul 29, 2026

Field Codes for Distributed Coupling Samplers and Certified Empirical Transport

In this paper, we formulate three communication tasks for empirical optimal transport: distributed coupling sampling, cost-evaluable coupling output, and scalar value-certified sampling. Our main result is a field-code compiler: any communicated transport field approximating an optimal empirical Monge map to error ηη can be completed by sparse target-cell residuals into an exact-marginal value-certified sampler with scalar certificate W1(μ,ν)≤U≤W1(μ,ν)+2ΔW_1(μ,ν)\leq U\leq W_1(μ,ν)+2Δ, where ΔΔ is the public target-partition diameter. The certificate accuracy is controlled by ΔΔ alone. The field error ηη controls residual communication under a cell-margin condition; without a margin, ηη alone does not bound residuals. We instantiate the compiler via adaptive local-affine and tensor-product spline codes with d(m+1)dbd(m+1)^db field bits in the spline case, plus residual lists charged separately. For lower bounds, exact Gap-Hamming embeddings prove certified output is hard, including a smooth cell-packing diffeomorphism family requiring Ω(ε−2d/(d+4))Ω(\varepsilon^{-2d/(d+4)}) communication for any cost-evaluable, cost-certified, or value-certified protocol. The same gadgets admit zero-communication samplers, formally separating the sampler and certificate-bearing output models. These results identify the transport field as the right communicated object whenever a field code is available, primarily as a residual-sparsity tool.

Hung Mai, Hai Nguyen, Luong Doan +4

1B0Labs, N2TP Technology · 3Nanyang Technological University · 4Phenikaa University, Vietnam

Communication ComplexityOptimal Transport
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Jul 29, 2026

FPSGen: Flexible Point Cloud Scene Generation with BEV-Supported Transport Flows

Existing point-based generative methods for outdoor scenes primarily focus on LiDAR-conditioned completion. During training, noisy point clouds are constructed by perturbing complete ground-truth scenes, whereas during inference, they are initialized by adding noise to duplicated partial scans. This train-inference mismatch inherits the sparsity and visibility bias of partial scans, leading to sparse distant regions and incomplete geometry in occluded areas. Moreover, the reliance on partial scans restricts generation when LiDAR observations are unavailable or replaced by layout cues. We present FPSGen, a flexible framework that constructs point sources independently of partial scans. FPSGen first predicts a bird's-eye-view (BEV) prior with density, height, and mask channels from the active cues. The density map is then sampled to form a BEV-supported point source, enabling both unconditional and conditioned initialization. A teacher-student approximate optimal transport scheme then uses teacher-predicted endpoints to learn a velocity field that induces straighter transport paths. By integrating BEV point source construction with path-straightening transport, FPSGen provides a unified framework for unconditional and flexible cue-conditioned scene generation. Extensive experiments show that FPSGen achieves state-of-the-art JSD and voxel IoU performance on SemanticKITTI completion while maintaining strong performance with a single point transport step. On KITTI-360 unconditional generation, it also achieves the best Coverage (COV) among the compared methods.

Wenzhe He, Meng Wang, JiaWei Qian +3

College of Computer Science and Electronic Engineering, Hunan University, Hunan, China · College of Information Science and Engineering, Hunan Normal University, Hunan, China

Point Cloud Completion3D Scene Generation3D Shape GenerationPoint Cloud GenerationOptimal Transport
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Jul 27, 2026

DrainSinkhorn: Safe Elimination for Batched Entropic Optimal Transport

Fast entropic optimal transport backends reduce the cost of each Sinkhorn update, but static batches still run at full width until the slowest problem finishes. We introduce DrainSinkhorn, a verifier-gated active-packing layer for batches of independent Sinkhorn problems. It combines candidate-axis packing, a Sinkhorn-specific one-sided screen, verifier-gated retirement under the backend's configured two-sided residual check, and physical compaction of all candidate-indexed state. The EOT objective, per-instance Sinkhorn map, and stopping rule are unchanged; later kernels run only on unfinished problems. We characterize the removable work exactly. If completion depths differ within a packed window, active execution removes the padding between the static batch rectangle and the observed survival curve. A quotient nonlinear Perron-Frobenius analysis gives a local explanation for these finite-tolerance depth differences: convergence depends on the full modal spectrum and proposal alignment, not only on the slowest mode. DrainSinkhorn achieves state-of-the-art execution performance on the tested heterogeneous batched-EOT workloads within matched backend families. The complete Flash-backed OT path is 4.110x faster on MetroPT-3, 3.798x faster on ImageNet-32 feature couplings, and 1.250-1.270x faster across a five-tolerance Packer19 sweep. Independent implementations reach 2.600x on ImageNet-32 with OTT-JAX, 3.174x on A2D2 LiDAR with PyKeOps, and 1.415x on large ImageNet-32 PyKeOps couplings. End-to-end speedups remain 4.074x on MetroPT-3 and 2.786x on ImageNet-32 feature-space OT flow matching, with all reported residual, consumer-output, and training-quality checks passing.

Xinyang Wen

GPU Kernel OptimizationOptimal Transport
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Jul 23, 2026

UnDA: Unpaired Domain Alignment for Cross-Modal Knowledge Transfer in Medical Imaging

Multimodal based approaches often outperform single modality approaches in downstream tasks as the different modalities provide complementary information, yet acquiring paired clinical data remains a significant challenge in real world scenarios. While cross-modal knowledge distillation addresses this, existing methods often struggle with large modality gaps and the propagation of noise from uncertain source-domain predictions. To overcome these challenges, we propose UnDA, an anchor-guided framework for unpaired cross-modal distillation. Our approach introduces a backbone-agnostic Alignment Module that extracts semantically structured class tokens via an attention based pooling mechanism. To ensure robust knowledge transfer, we propose Uncertainty-Weighted Optimal Transport (UCT-OT), which dynamically weights feature-level alignment based on prediction confidence, effectively suppressing noisy supervision. Furthermore, a per-class ProtoNCE objective maintains stable prototype memories to enforce global discriminability across unpaired batches. Evaluations on representative segmentation tasks under strictly unpaired settings show consistent improvements in accuracy and boundary precision in the target modality, demonstrating that meaningful structural knowledge can be transferred across heterogeneous data sources without paired datasets.

Rafsan Jany, Shadab Tanjeed Ahmad, Ahsan Bulbul +3

Korea Institute of Oriental Medicine, Daejeon, South Korea · Islamic University of Technology, Gazipur, Bangladesh

Cross-Modal Knowledge DistillationCross-Modality Medical Image SegmentationCross-Modal AlignmentOptimal Transport
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Jul 22, 2026

Attribution Markets: A Fisher-Market Formulation for Fractional Credit Assignment Between Planned Tasks and Performed Actions

Personal and organizational planning systems maintain two records that drift apart: what was planned (a task's effort budget) and what was done (a logged action's duration and description). Existing systems bridge them with an exclusive, all-or-nothing link that strands genuinely related but unlinked effort and reports false stalls on active goals. We formulate the bridge as a quasi-linear Fisher market: planned tasks are budget-constrained buyers, performed actions are divisible goods, and a fused text/structural/temporal signal sets each buyer's valuation. Two market instruments - a seller reserve price and a buyer cash option - yield conservation, a hard budget cap, and a provable junk filter as theorems. We extend the market with a concave completion utility discounting progress as a task nears its plan; standard convergence theory for the market's algorithm does not transfer here, resolved by a satiation-threshold fixed point with existence (Brouwer) and local uniqueness under an explicit diagonal-dominance condition, validated empirically on random and adversarial instances. A de-circularized, multi-seed benchmark - observed affinity corrupted independently of the scored ground truth - surfaces a genuine weak spot: the market's sharp, zero-entropy equilibrium is more sensitive to affinity noise than entropy-regularized optimal transport's permanently smoothed one. We resolve this with a one-parameter entropy-regularized generalization unifying the two, plus a noise-adaptive rule for its regularization strength. We report full reproducibility parameters, discuss limitations candidly, and relate the result to multi-touch attribution, optimal transport, and online Fisher-market algorithms.

Salavat Ishbulatov

Independent researcher

Mechanism DesignFine-Grained Credit AssignmentOptimal Transport
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Jul 19, 2026

OTAP: Structure-Aware Optimal Transport for Evaluating Planning and Execution in Agent Trajectories

Large language model agents solve tasks by generating trajectories that interleave planning, tool calls, and intermediate results. Current evaluation metrics reduce such a trajectory to a binary success flag, compare it against a reference by exact matching, or delegate judgment to another language model. A success flag cannot distinguish a sound solution from one that succeeds by luck, and says nothing about why a failed run went wrong. Exact matching penalizes plans that are valid but reordered or decomposed differently from the reference. We reframe trajectory evaluation as a distance between the agent's execution graph and a set of valid solution graphs, and instantiate it via an unbalanced fused Gromov-Wasserstein transport problem over attributed dependency graphs. The resulting score, termed OTAP (Optimal Transport for Agentic Planning), is a pseudo-metric that is provably invariant to dependency-preserving reorderings and has bounded sensitivity to redundant steps. Its unbalanced marginals handle missing or hallucinated steps without forcing a match, and its soft coupling accommodates variation in plan granularity. On controlled perturbations and three public benchmarks, OTAP separates valid from invalid trajectories in a regime where semantics-only metrics score below chance. Its advantage tracks the fidelity of the dependency graph: largest where edges follow from operator semantics, smallest where they are inferred from free text. Where a formal verifier exists, strict surface metrics predict validity better than OTAP does, which places OTAP in open-ended domains where no verifier is available.

Babak Barazandeh, Subhabrata Majumdar, George Michailidis

Fortinet · IIMB · UCLA

LLM Agent EvaluationAI Agent EvaluationAgent Trajectory AnalysisOptimal Transport
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Jul 18, 2026

Twisted Schrödinger Bridge Matching

Over the past few years, diffusion-based Schrödinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schrödinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schrödinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schrödinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous- and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schrödinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data. Code available at https://github.com/maxencenoble/twisted-sb-matching.

Maxence Noble, Marie Scheid, Yazid Janati +2

CMAP, Ecole polytechnique · Institute of Foundation Models · MBZUAI +1

Schrödinger BridgesTrajectory InferenceOptimal TransportDiffusion Models
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Jul 17, 2026

Cluster-Aware Matching via Laplacian Optimal Transport

In many applications of matching, the point clouds to be matched are not merely unstructured sets of points but rather samples from distributions with an intrinsic cluster structure. In such cases, as individual points are often interchangeable within a coherent region, finding a robust region-to-region alignment is more desirable than establishing a precise point-to-point correspondence. To this end, we propose a novel approach for cluster-aware matching based on Laplacian Optimal Transport (LapOT). The key idea is to regularize the optimal transport problem with quadratic Laplacian terms constructed from similarity graphs of the point clouds, which encourages the optimal coupling to respect the cluster structure of both point sets. We also introduce Refined Simultaneous Clustering (RSC), a method that leverages the cluster-aware coupling obtained from LapOT to produce consistent partitions across the point sets, which can overcome the limitations of independent clustering and yield more stable and interpretable results. We demonstrate the effectiveness of our approach through theoretical analysis and empirical experiments, showing that LapOT indeed produces cluster-aware matching that leads to more consistent and meaningful alignments between point clouds.

Gabriel Samberg, YoonHaeng Hur, Yuehaw Khoo +1

Department of Applied Mathematics, Tel Aviv University · Department of Statistics, Columbia University · Department of Statistics, University of Chicago

ClusteringGraph MatchingPoint Cloud RegistrationOptimal Transport
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Jul 12, 2026

Fast Whole-Brain, Geometry-Aware Functional Alignment for Cross-Subject Decoding

Decoding brain activity is useful for characterizing brain processes and understanding the functional architecture underlying cognition. However, the inter-individual variability in brain response patterns limits the development of decoders that generalize across individuals. A solution to this challenge is functional alignment: aligning functional data across individuals before training population-level decoders. The core issue is to strike the balance between aligning functional features and preserving the anatomical structure, while maintaining computational efficiency. We introduce a new functional alignment method for fMRI, SpectralOT, that embeds cortical geometry into Laplace-Beltrami eigenmodes along functional data to regularize the alignment.

Pierre-Louis Barbarant, Florent Meyniel, Bertrand Thirion

Université Paris-Saclay, Inria, CEA, Palaiseau 91120, France · Cognitive Neuroimaging Unit, Institut National de la Santé et de la Recherche Médicale, Commissariat à l’Energie Atomique et aux énergies alternatives, Université Paris-Saclay, NeuroSpin center, 91191 Gif/Yvette, France · Institut de neuromodulation, GHU Paris, psychiatrie et neurosciences, centre hospitalier Sainte-Anne, pôle hospitalo-universitaire 15, Université Paris Cité, Paris, France

Neural DecodingNeuroimagingRepresentation AlignmentOptimal Transport
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Jul 12, 2026

Beyond Looking Up, Try Looking Around: Harmonizing Global Structure and Local Consistency in Optimal Transport for Short Text Clustering

Pseudo-labeling based on Optimal Transport (OT) has become an effective mechanism for enhancing short text clustering. Existing OT methods are short in modeling semantic consistencies between samples, which may assign different pseudo-labels to semantically similar samples. These erroneous pseudo-labels can cause the model to produce inferior clusters. This paper proposes a novel short text clustering framework, which remedies the neglect of semantic consistency in existing OT methods, generating reliable pseudo-labels to facilitate clustering. Specifically, the proposed approach first designs an instance-level attention mechanism to capture semantic relationships between samples, which are then integrated into the OT formulation to endow the transport process with neighborhood semantic awareness. By solving the proposed OT formulation, reliable pseudo-labels are obtained that simultaneously account for sample-to-sample semantic consistency and sample-to-cluster global structure information. These pseudo-labels are then used as supervisory signals to guide the model to achieve accurate clustering. Extensive experiments demonstrate that the proposed approach outperforms state-of-the-art methods. The code is available at: https://github.com/YZH0905/CAOT-STC.

Zhihao Yao, Yuxuan Gu, Jixuan Yin +1

Harbin Engineering University, China · Harbin Institute of Technology, China

ClusteringUnsupervised ClusteringPseudo-LabelingOptimal TransportText Clustering
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Jul 11, 2026

Scalable Optimal Transport Algorithm for Network Alignment

Network alignment identifies node correspondences across different networks and is a fundamental primitive in many data science applications, including social network analysis, fraud detection, and knowledge graph integration. However, state-of-the-art network alignment methods often achieve high accuracy by repeatedly constructing and updating dense matrices, sacrificing scalability in the process. To address this scalability limitation without compromising alignment accuracy, we present FastAlign, a scalable, sparsity-aware framework for optimal transport-based network alignment. Rather than introducing a new alignment model, FastAlign preserves the original OT formulation and reinterprets its computation as a set of recurring mixed sparse-dense operations. FastAlign combines sparsity-aware graph computation with domain-specific kernel fusion, including a custom SpMM kernel. Our results show that FastAlign achieves alignment quality comparable to state-of-the-art OT-based methods while substantially reducing end-to-end runtime up to 3.89x-9.45x on CPU and 2.24x-32.54x on GPU.

Elaheh Hassani, Durga Mandarapu, Qi Yu +2

Lawrence Berkeley National Laboratory CA, USA · University of Illinois at Urbana-Champaign IL, USA

Graph MatchingSparse Matrix MultiplicationOptimal Transport
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Jul 10, 2026

Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems

We study the optimal transport of optimally controlled agents from a compactly supported absolutely continuous source to a discrete target measure. The ground cost for the transport is induced by the optimal cost of the agents' motion. When this ground cost satisfies the twist condition, the optimal transport map is given almost everywhere in terms of a Laguerre tessellation of the state space. We refer to this control-theoretic generalization of Laguerre tessellation as Control Laguerre Tessellation (CLT), and illustrate it for two ground costs induced by linear controlled agents with minimum energy and minimum time objectives.

Ripon C. Sarker, Abhishek Halder

Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA

Optimal ControlOptimal Transport
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Jul 6, 2026

Wasserstein Residuals: Learning Gradient Flows from Population Dynamics

Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan--Kinderlehrer--Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code see github.com/BasisResearch/wasserstein-residuals.

Markus Heinonen, Yair Shenfeld, Ricardo Baptista +4

Basis Research Institute · Aalto University · Brown University +2

Dynamical SystemsWasserstein Gradient FlowsResidual LearningTrajectory InferenceOptimal Transport
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Jun 29, 2026

The Fundamental Limits of Valid Transport Map Estimation

Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.

Sivaraman Balakrishnan

Department of Statistics and Data Science† · Machine Learning Department‡

Minimax EstimationGenerative ModelingOptimal Transport
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Jun 29, 2026

ITSPACE: Monotone Gaussian Optimal Transport Updates

Covariance matrices serve as compact descriptors of feature distributions in many machine-learning pipelines, including domain adaptation and Gaussian embeddings. Under a centered Gaussian approximation, the unregularized Wasserstein-2 optimal-transport (OT) discrepancy admits a closed form on covariances given by the Bures-Wasserstein (BW) objective on the symmetric positive definite (SPD) cone. We propose ITSPACE (Iterative Transport for Stable Proximal Alignment of Covariance Embeddings), a proximal majorization-minimization method that directly optimizes this exact BW objective through closed-form updates in a square-root factorization. In exact arithmetic, each iteration satisfies a sufficient-decrease inequality for the BW objective; under inexact polar computations, we provide an explicit certificate-gap bound controlling deviations from exact descent. The resulting iterations preserve PSD structure by construction and naturally support rank-restricted factors, making ITSPACE well-suited as a lightweight inner-loop primitive in settings where adaptation must be performed from unlabeled target batches under strict step and compute budgets. Across real-world covariance-alignment benchmarks, ITSPACE reaches low-BW-gap solutions substantially faster than BW-gradient descent, methods based on other covariance geometries, and entropically regularized sample-OT baselines.

Woojoo Na, Jennifer Dy

Department of Computer Engineering, Northeastern University, Boston, USA.

Matrix OptimizationOptimal Transport
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Jun 29, 2026

Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions

The Sliced Wasserstein (SW) distance has emerged as a computationally attractive alternative to the Wasserstein distance by leveraging one-dimensional optimal transport along random projections. Standard estimators of the SW distance rely on Monte Carlo averages of one-dimensional Wasserstein distances computed via quantile functions, which require sorting projected samples and access to full datasets. In this work, we introduce a new class of estimators for the Sliced Wasserstein distance based on cumulative distribution functions (CDFs) of projected measures, that avoid sorting and scale via massive dataset parallelism. This class includes several estimators, some of them being indexed by hyperparameters controlling their variance or smoothness. We show that they are especially well suited to scenarios in which CDFs are more tractable than quantile functions, such as mixtures of Gaussians, and moreover that they are also naturally compatible with federated learning, since CDFs of projected data can be computed and aggregated locally without requiring the exchange of raw samples.

Christophe Vauthier, Quentin Mérigot, Anna Korba

Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay Gif-sur-Yvette, France · Centre de recherche en économie et statistique, ENSAE Palaiseau, France

Sliced Wasserstein DistanceOptimal TransportWasserstein Distance
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Jun 29, 2026

Notes on generative modeling: flow matching, diffusion, optimal transport and Schr{ö}dinger bridge

These notes recapitulate the high level mathematical principles behind different techniques for generative modeling. I show the connections between optimal transport and standard techniques such as Schr{ö}dinger bridge and flow matching.

Titouan Vayer

COMPACT · Inria, Rennes, France.

Flow MatchingSchrödinger BridgesGenerative ModelingFlow-Based Generative ModelingOptimal TransportDiffusion Models
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Jun 28, 2026

MIRROR: Aligning Semantic Relations from Language to Image via Gromov--Wasserstein

Multimodal Large Language Models (MLLMs) inherit rich relational priors from their language backbones, yet often fail when asked to apply these relationships in visual contexts. We trace this failure to a structural blind spot: projection-based alignment trains each visual token to carry the right semantics, but never asks whether the relationships between concepts survive the crossing from language to vision. To address this, we propose MIRROR (Mapping Inter-concept Relations from language to visual Representation via Optimal-transport-based Regularization), a geometric regularization framework that transfers relational priors from language to vision by exploiting the rich relational structure encoded in language representations. Specifically, we derive a surrogate loss from the proposed Semi-Inverse Gromov-Wasserstein (SI-GW) problem, an inverse geometric problem that aligns visual representations with language-derived relational priors. We show that this formulation admits a unique closed-form solution that prescribes the ideal visual relational structure implied by language geometry and cross-modal coupling. The structure of the formulation also enables efficient computation, making it applicable to long token sequences. Applying SI-GW inside decoder-only Transformers requires careful design. We introduce targeted strategies at the layer, head, and token levels to ensure stable extraction without additional parameters or inference cost. MIRROR improves relational consistency while preserving performance on general vision-language tasks.

Hong-Han Wang, Yuntao Wang, Hu Ding

University of Science and Technology of China, Hefei, China

Large Vision-Language ModelsVLM ReasoningModality Gap in VLMsMultimodal Foundation ModelsVision-Language AlignmentOptimal Transport+1
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Jun 26, 2026

Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization

Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.

Clément Bonet, Pierre-Cyril Aubin-Frankowski, Youssef Mroueh

CMAP, CNRS, Ecole Polytechnique, IP Paris · CERMICS, CNRS, ENPC, IP Paris · IBM Research

Maximum Mean DiscrepancyNonconvex OptimizationOptimal Transport
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Jun 25, 2026

Bridging Vision and Language Concepts through Optimal Transport Semantic Flow

Concept Bottleneck Models (CBMs) promise transparent reasoning by predicting through human-interpretable concepts, yet their effectiveness fundamentally depends on how well visual and textual representations are aligned or matched. Existing vision-language CBMs often rely on pre-aligned encoders or global cosine similarity, which obscures fine-grained concept localization and fails to reflect true semantic geometry. In this work, we rethink concept alignment as a dynamic cross-modal transport process instead of static projection and propose the Optimal Transport Flow Concept Bottleneck Model (OTF-CBM). It first learns a data-driven semantic cost via Inverse Optimal Transport to measure cross-modal distances, and then performs unbalanced optimal-transport-based flow matching to model semantic transitions between visual patches and textual concepts. With velocity-based concept activation, OTF-CBM captures interpretable geometric relations without ODE integration. Experiments further show that OTF-CBM achieves superior classification accuracy and concept faithfulness, offering a new geometric and dynamical perspective for interpretable cross-modal reasoning.

Chenyang Zhang, Anqi Dong, Guangming Zhu +4

School of Computer Science and Technology, Xidian University, Xi’an, China · KTH Royal Institute of Technology, Stockholm, Sweden

Concept Bottleneck ModelsVLM InterpretabilityVision-Language AlignmentOptimal TransportMultimodal Understanding
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Jun 24, 2026

Variational Inference via Entropic Transport Descent

Particle-based variational inference (ParVI) methods approximate an intractable target distribution by evolving an ensemble of interacting samples. Existing approaches rely predominantly on kernel-based repulsion (e.g., SVGD), which suffers from variance collapse in high dimensions and mode collapse on multimodal targets -- pathologies caused by the absence of global transport structure. We introduce entropic transport descent (ETD), a ParVI family that frames each particle update as an entropy-regularized optimal transport problem. Derived from the JKO proximal scheme by lifting to the space of couplings and relaxing via the KL chain rule, each ETD iteration reduces to a Sinkhorn computation. The resulting transport plan provides global coordination, guiding each particle to nearby high-density proposals and naturally preserving multimodal structure. ETD can operate entirely score-free, requiring only pointwise evaluations of the unnormalized target density. Experiments on variance-collapse diagnostics, Bayesian logistic regression, neural networks, and molecular Boltzmann distributions show that ETD matches or outperforms SVGD, AGF-SVGD, and SGLD, with the largest gains in high-dimensional and multimodal settings.

Vincent Pacelli, Akash Ratheesh, Evangelos Theodorou

School of Aerospace Engineering Georgia Institute of Technology Atlanta, GA 30332

Bayesian InferenceVariational InferenceOptimal Transport
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Jun 23, 2026

Counting Trees from Satellite Imagery with Noisy Supervision

Counting individual trees is a fundamental task for environmental monitoring, yet remains largely unexplored with satellite imagery. At these resolutions, isolated trees may still be identifiable, but crown boundaries become ambiguous in dense forests, making the notion of an individual tree inherently ill-defined. Moreover, large-scale manual annotations of individual trees are prohibitively expensive. While scalable supervision can be derived from airborne LiDAR, the resulting annotations are noisy and difficult to exploit effectively. We address these challenges by formulating tree counting as a spatial density matching problem supervised through Unbalanced Optimal Transport. This formulation naturally accommodates both precise localization of isolate trees and robust density estimation in dense forests. We further introduce a self-correction mechanism that leverages transport residuals to progressively refine noisy supervision during training. We evaluate our approach on TinyTrees, a new benchmark spanning three continents and three satellite sensors, comprising over 216 million tree annotations (including 639k manually verified instances) across 25 89025\,890 km2^2. Our method consistently outperforms detection-based, regression-based, and transport-based distribution-matching baselines, demonstrating the effectiveness of unbalanced transport and reliability-aware supervision for large-scale tree counting from satellite imagery. Code, data and models are available at https://github.com/dgominski/treematch.

Dimitri Gominski, Maurice Mugabowindekwe, Qiue Xu +6

University of Copenhagen · University of Rwanda · University of Chinese Academy of Sciences +3

Remote SensingLearning with Noisy LabelsRemote Sensing BenchmarksOptimal Transport
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Jun 23, 2026

Sample complexity of unbalanced entropic OT

Optimal transport (OT) has become a central language for comparing probability measures, but exact balanced OT is often both too rigid for data with missing, created, or destroyed mass and subject to unfavorable high-dimensional sample complexity. Entropic regularization and unbalanced relaxations address these limitations in complementary ways. Entropy smooths the geometry, improves statistical behavior, and enables fast Sinkhorn-type algorithms, while unbalanced marginal penalties replace hard conservation constraints by divergence terms adapted to noisy empirical data. This paper studies the sample complexity of entropic unbalanced OT at the level of the optimal coupling, rather than only the scalar transport value. We develop a translation-invariant dual formulation, prove compactness and strong convexity properties for the intrinsic dual variables, and convert these geometric estimates into high-probability finite-sample bounds for empirical couplings. The results clarify why regularization is a practical necessity in machine learning applications: it softens the curse of dimensionality, reduces the number of samples needed for stable transport estimation, and keeps the resulting estimators compatible with scalable Sinkhorn-type solvers.

Francisco Andrade, Gabriel Peyré, Clarice Poon

INRIA (PreMeDICaL & HeKA) · INRIA · CNRS and ENS, PSL Universit´e +1

Sample ComplexityEntropy RegularizationOptimal Transport
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Jun 23, 2026

The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space

The space P2(Rd\mathcal{P}_2(\mathbb{R}^d) of probability measures with finite second moment carries a natural geometry: the quadratic Wasserstein distance W_2 makes it a complete metric space and, following Otto, a (formal) Riemannian manifold whose geodesics are the optimal-transport interpolations. On this manifold, the gradient flow of the free energy F(rho) = KL(rho || π) is exactly the Fokker-Planck equation, and its implicit-Euler discretization is the JKO scheme. This is the geometry underlying diffusion models: the forward process descends the free energy, and each denoising step realizes one JKO step, which recovers DDPM, DDIM, NCSN/SMLD, and Energy Matching; this is one scheme, not separate theories. The same manifold supports a second variational principle. Its geodesics - the minimum-action curves of the Benamou-Brenier formula - are precisely the optimal-transport paths that Flow Matching learns. Fixing both endpoints and following the geodesic, generation becomes a deterministic ODE along a straight line, hence far fewer sampling steps. Placing both families of models on one manifold makes their relationship exact: diffusion follows a free-energy gradient flow, an initial-value problem; optimal-transport Flow Matching follows a Wasserstein geodesic, a boundary-value problem. The two reach the same endpoints along different paths.

Yian Yao, Weiwei Zhang

from the mortal world · from heaven

Flow MatchingWasserstein Gradient FlowsOptimal TransportDiffusion Models
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Jun 19, 2026

A Transport-Based Geometry of Belief-Cost

A finite agent, a machine's digital twin or any bounded reasoner, infers a fixed and noisy world through finite sensors, so its coherent output is a belief: a probability density over states (the Bayes posterior). Such an agent stops short of certainty, and revising a belief carries a cost. We propose a framework for belief costs based on optimal transport, motivated by these facts. We pose two postulates. P0 (the arena): a revision cost is a scalar price on optimal transport, so beliefs live in Wasserstein space. P1 (uniform pricing): one nat of knowledge costs the same metric length everywhere, the eikonal condition. Among conceivable pricing rules we study this one. Under P0 and P1 the cost metric is optimal transport conformally reweighted by Fisher information, g~e,U=2(e+U) gW2\tilde g_{e,U}=2(e+U)\,g_{W_2}, and the Fisher family is a characterization: among continuous reliefs, uniform pricing is equivalent to U=cJU=cJ. Two consequences follow on the conformal class. Certainty sits at infinite cost-distance once the relief dominates the Fisher information, so a well-posed inference has a cost floor diverging at certainty (necessity conjectural beyond power laws). On location-scale leaves the geometry is hyperbolic, and the Stam bound places the Gaussian as the most curved one (at e=0e=0). The results are geometric, in nats, and hold up to units: a change of cost unit rescales all distances and preserves every conclusion (boundary, eikonal family, hyperbolicity, Gaussian extremum), a gauge theorem; a global change of state units at e=0e=0 is an isometry; the content lies in signs, rankings and ratios. Via Landauer (one nat worth kBTk_BT) the cost floor becomes an energy floor: revising toward certainty would demand unbounded energy. Physics anchors the unit and enters no theorem. Removing either postulate leaves the selection open.

Laurent Caraffa

Université Gustave Eiffel, LASTIG, IGN-ENSG

Information GeometryOptimal TransportWasserstein Distance
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Jun 19, 2026

Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transport

Quantum pure-state ensembles live on complex projective space, making flat Euclidean generative modeling geometrically mismatched. We introduce Intrinsic Flow Matching (IFM), a deterministic transport framework on CPd−1\mathbb{CP}^{d-1} that learns tangent velocity fields using Pancharatnam phase-aligned conditional paths. IFM replaces local score teachers and reverse-time stochastic sampling with manifold probability flow, while horizontal parameterization removes redundant ambient directions. We show that the IFM objective recovers the induced marginal transport field, represents deterministic projective ensemble flows, and yields endpoint and stability guarantees. Empirically, IFM often improves over ambient Euclidean flow matching across higher-qubit, multimodal, spin-coherent, physics-inspired, and amplitude-encoded MNIST image-vector benchmarks, with strongest gains on high-dimensional and coherence-sensitive tasks but not uniformly across every metric.

Jian Xu, Delu Zeng, John Paisley +1

RIKEN iTHEMS · RIKEN AIP · South China University of Technology +1

Flow MatchingQuantum Generative ModelingQuantum Machine LearningOptimal Transport
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Jun 18, 2026

OTCHA: Optimal Transport-driven Confidence-aware Latent Hub Alignment for Multi-View Medical Image Classification

Multi-view imaging, such as mammography and chest radiography, is a standard component of clinical practice. However, medical images are often unregistered and contain view-specific artifacts or irrelevant background cues that can obscure diagnostically relevant findings. Many existing methods directly fuse per-view representations, allowing such irrelevant content to contaminate the fused embedding and reducing robustness under varying view configurations. We propose OTCHA, a confidence-aware latent hub token alignment module based on optimal transport (OT) that refines patch tokens before fusion for multi-view classification. OTCHA introduces a set of learnable latent hub tokens shared across views. For each view, we compute an OT plan between patch tokens and hub tokens that jointly considers feature similarity and geometry, and augment the OT formulation with token-conditional dustbins to enable partial matching and discard irrelevant tokens. The resulting transport plan provides token-wise matching confidence, which gates hub-mediated message passing and weights a novel optimal-transport-based representation alignment loss to stabilize refinement. Experiments on three multi-view medical image datasets demonstrate consistent improvements over competing baselines across diverse anatomies and view configurations. Our code is available at https://github.com/labhai/OTCHA.

Jiwoong Yang, Haejun Chung, Ikbeom Jang

Hanyang University, Seoul, Republic of Korea · Hankuk University of Foreign Studies, Yongin, Republic of Korea

Multi-View ConsistencyMulti-View LearningMedical Image ClassificationOptimal Transport
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