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

Jun 15, 2026

Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence

This paper is concerned with learning principal variations of random probability measures on

\mathbb{R}^m

under the Wasserstein geometry. We introduce a new dynamical formulation to interpret the log-PCA, a linearized principal geodesic analysis, as a variational approach. Our differentiable version, termed as the Wasserstein Tangential PCA (WT-PCA), captures the local principal modes of geodesic variations of a (weighted) probability measure on the Wasserstein space via its covariance operator at barycenter. Based on the dynamical perspective and leveraging parallel transport structure of the optimal transport problems, we derive a general statistical convergence rate of the empirical WT-PCA when estimated from data in terms of the 2-Wasserstein distance between the population and empirical barycenter reference measures.

Peng Xu, Changbo Zhu, Young-Heon Kim +1

Department of Statistics University of Illinois Urbana-Champaign · Department of ACMS University of Notre Dame · Department of Mathematics University of British Columbia +1

Principal Component Analysis Statistical Learning Theory Optimal Transport

Jun 15, 2026

Sinkhorn-CPD: Robust point cloud registration via unbalanced entropic optimal transport

Coherent Point Drift (CPD) is widely used for rigid point cloud registration because of its soft correspondences and closed-form parameter updates. However, CPD's target-side marginal constraint forces every observation, including outliers, to receive exactly unit probability mass. This assumption degrades registration accuracy under heavy outliers and partial overlap. Optimal transport (OT) methods can handle missing mass through unbalanced formulations, but require hand-tuned annealing schedules. In this paper, we propose Sinkhorn-CPD, which replaces CPD's target-side marginal constraint with dual Kullback-Leibler penalties, allowing the algorithm to discard outliers on both sides. The resulting formulation is a fully unbalanced entropic optimal transport problem, which can be efficiently solved by generalized Sinkhorn iterations. Moreover, Sinkhorn-CPD preserves the closed-form Procrustes and variance updates of CPD. In our method, the variance sigma^2 plays the role of the entropic regularization parameter, which induces an automatic annealing schedule from diffuse to sharp correspondences without manual temperature tuning. Experiments on synthetic, cross-category, and scan-to-CAD benchmarks show that Sinkhorn-CPD achieves state-of-the-art accuracy, with strong robustness to outliers and partial overlap.

Jin Zhang, Mingyang Zhao, Bing Liu +1

LMIB & School of Mathematical Sciences, Beihang University, Beijing, China · State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China · University of Chinese Academy of Sciences, Beijing, China +1

Point Cloud Registration Optimal Transport

Jun 15, 2026

Distribution Alignment for One-Shot Federated Learning via Optimal Transport

One-Shot Federated Learning (OSFL) addresses extreme communication regimes in which clients interact with the server only once, amplifying the impact of heterogeneous client data distributions. In particular, the interaction of domain shift and label shift across clients induces misaligned feature representations that cannot be corrected through iterative optimization. Existing OSFL methods rely on distillation, server-side generation or ensemble-based aggregation, but assume aligned representations or address domain and label shift separately. We introduce SLOT-Align (Single-round, Learning-free Optimal Transport Alignment), a geometry-aware feature harmonization framework for OSFL. SLOT-Align uses a shared frozen encoder to extract compact feature statistics, constructs a global reference via Bures-Wasserstein barycenters, and aligns local representations using closed-form geodesic optimal transport maps. The method is computationally efficient and can be combined with existing OSFL pipelines relying on frozen encoders without modifying their training procedures. Extensive experiments across multiple benchmarks, pretrained backbones, and OSFL methods show that SLOT-Align consistently improves accuracy and robustness under joint domain and label shift.

Daniele Berardini, Vito Paolo Pastore, Vittorio Murino

AI for Good · AI for Good (AIGO), Italian Institute of Technology, Genoa, Italy · MaLGa-DIBRIS, University of Genoa, Genoa, Italy +1

Distribution Matching Optimal Transport Federated Learning Heterogeneous Federated Learning

Jun 15, 2026

Generative Modeling on Metric Graphs via Neural Optimal Transport

We introduce, to our knowledge, the first deep generative modeling framework for probability distributions continuously supported on compact metric graphs. Given source and target measures on a metric graph, our method embeds the graph into a smooth ambient space, solves an entropic Kantorovich problem via a neural semidual parameterization, and projects generated samples back onto the original graph. We study two embedded geometries: an extrinsic Euclidean realization and the intrinsic tropical Abel--Jacobi embedding into the Jacobian torus. In both cases, the resulting generator is graph-supported by construction. We prove that, in the joint limit of increasing neural expressivity, the learned generator converges weakly to a valid transport coupling between the original graph measures. Empirically, across a range of geometrically distinct graphs, our method matches or improves upon heuristic transport baselines based on discrete graph OT, while scaling more favorably. Finally, we demonstrate scalability on real-world urban mobility data by training our model on one million Uber pickup locations in Manhattan, New York City.

Alessandro Micheli, Yueqi Cao, Anthea Monod +1

Imperial College London London, UK · KTH Royal Institute of Technology Stockholm, Sweden · Statens Serum Institut Copenhagen, Denmark +1

Generative Modeling Optimal Transport

Jun 12, 2026

Learning Urban Access Costs from Origin-Destination Flows via Inverse Optimal Transport

Cities deliver basic services through mixed public-private facility networks, including schools, clinics, transit providers, and subsidized service points. In these systems, planners often observe where households go, but not the latent cost function through which they trade off factors such as distance, price, and institutional access. We study this urban problem through school choice in the Philippines, where the country's largest national education subsidy is intended to redirect learners from congested public schools to participating private schools. Treating school-to-school enrollment flows as an entropic optimal transport plan, we recover latent choice costs using two complementary inverse optimal transport models: an interpretable distance-banded model with a subsidy term, and a neural cost model trained through a differentiable Sinkhorn forward pass. Applied to 283{,}016 learner trips across 23{,}820 observed flows in the most populated region, the framework estimates a subsidy-equivalent distance,

λ^{(k)}

, interpreted as the kilometers of perceived travel cost offset by the subsidy. The case demonstrates how administrative origin-destination data can be transformed into interpretable planning metrics for accessibility-aware subsidy design, facility siting, and urban service allocation.

Paula Joy B. Martinez, Sebastian Felipe R. Bundoc

Inverse Optimal Transport Urban Planning Optimal Transport

Jun 12, 2026

Geometric Domain Adaptation via Optimal Transport for Linear Regression in R^2

Optimal Transport has become recently a powerful method for domain adaptation by aligning source and target distributions. We study a supervised domain adaptation problem where source and target domains are related by a rotation or a translation or a homothety in

\mathbb{R}^2

. We prove that the optimal transport map recovers the underlying map when using a

p-

norm cost with

p \geq 2

. Based on this insight, we develop a method combining

K-

means and optimal transport to estimate the underlying map, enabling adaptation of linear regression models when target data is scarce. Simulations demonstrate improved performance over baseline methods. Rather than relying on highly expressive deep learning architectures, we focus on classical machine learning models to emphasize interpretability and theoretical insight. This perspective allows us to explicitly characterize the role of optimal transport in recovering geometric transformations such as rotations, translations, and homotheties. Our contributions include a theoretical result linking optimal transport and rotations, translations and homothecies in

\mathbb{R}^2

, and a practical method for adaptation in linear regression offering both conceptual clarity and applied value in domain adaptation tasks in this space.

Brian Britos, Mathias Bourel

IESTA, Facultad de Ciencias Económicas y de Administración, Universidad de la República, Montevideo, Uruguay · IRL-2030, Instituto Franco-Uruguayo de Matemática e Interacciones (IFUMI)

Domain Adaptation Linear Regression Optimal Transport

Jun 11, 2026

Layer-Resolved Optimal Transport for Hallucination Detection in NMT and Abstractive Summarization

Optimal transport (OT) has been shown to detect hallucinations in neural machine translation (NMT) by measuring the geometric distance between cross-attention distributions and a reference distribution, without any supervision. We extend this analysis to all six decoder layers of the Fairseq DE-EN model (

N=3{,}414

), showing that Wass-to-Unif and Wass-to-Data are complementary detectors specialised across hallucination types, that detection is concentrated in layers L1--L4 with L5 anti-predictive for subtler types, and that hallucinated translations lack the exploratory attention phase present in correct translations from the first decoding step. We further evaluate whether the geometric signal transfers to abstractive summarization faithfulness detection: our unsupervised OT detector on AggreFact (

N=1{,}116

) achieves

57.2\%

/

57.6\%

balanced accuracy on CNN/XSum -- above chance but substantially below supervised MiniCheck-Flan-T5-L(

69.9\%

/

74.3\%

). This gap is principled: unlike NMT hallucinations, unfaithful summaries can attend correctly to source tokens while misrepresenting their content, a failure mode invisible to concentration-based OT metrics by construction. Structural experiments on T5-base confirm consistent decoder organisation across depth, with Layer~~3 showing peak concentration and Layer~~12 being most critical for generation quality. Together, the results establish OT on cross-attention as a reliable detector when the failure mode is source disengagement, a principled interpretability tool regardless of task, and fundamentally limited when faithfulness failures occur downstream of attention.

Mariia Onyshchuk, Maksym-Vasyl Tarnavskyi, Marta Sumyk

Faculty of Applied Sciences, Ukrainian Catholic University, Lviv, Ukraine.

Transformer Interpretability Hallucination Detection Factual Consistency Evaluation Optimal Transport

Jun 10, 2026

A Riemannian Approach to Low-Rank Optimal Transport

Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape's curvature. To address these limitations, we propose a unified Riemannian geometric framework for low-rank OT, modeling balanced and unbalanced rank-

r

positive factored couplings as novel smooth embedded submanifolds of the positive orthant. By equipping these manifolds with the Fisher-Rao product metric, we derive tractable formulations for Riemannian projectors, retractions, and Hessian-vector products. Our cost-agnostic framework seamlessly extends to linear OT, Gromov-Wasserstein (GW), fused GW, and their unbalanced counterparts. For balanced OT, our geometric ingredients are computed via efficient conjugate-gradient and iterative Bregman updates. For the unbalanced OT, our operations elegantly reduce to closed-form scalings, completely eliminating inner iterative loops. In both regimes, per-iteration complexity scales linearly with dataset size, and we provide a rank-sufficiency certificate for global optimality verification. Extensive experiments across a range of problem sizes demonstrate that our regularization-free first- and second-order solvers achieve faster convergence and superior performance over existing state-of-the-art low-rank OT solvers.

Pratik Jawanpuria, Bamdev Mishra

Centre for Machine Intelligence and Data Science, IIT Bombay · Microsoft India

Riemannian Optimization Second-Order Optimization Optimal Transport

Jun 8, 2026

A Source Domain is All You Need: Source-Only Cross-OS Transfer Learning for APT Anomaly Detection via Semantic Alignment and Optimal Transport

Advanced Persistent Threats (APTs) are stealthy, multi-stage cyberattacks whose detection is difficult due to scarce labeled traces, severe class imbalance, and the challenge of generating realistic malicious behavior. These challenges are amplified in cross-operating-system (cross-OS) settings, where a detector trained on one source platform must be deployed on an unlabeled target platform without access to target-domain labels. We study this source-only cross-OS APT detection problem using system-level provenance traces and propose a transport-based framework for ranking anomalous target processes under zero target supervision. The framework abstracts process behavior into structured natural-language descriptions, embeds them using pretrained language models, and constructs a source-normal reference for target scoring. It combines three evidence channels: semantic deviation from source-normal prototypes, structural deviation captured by graph autoencoding, and geometric deviation measured through Optimal Transport (OT). The main contribution is an OT-based barycentric anomaly score that projects target embeddings onto the source-normal manifold and quantifies residual transport mismatch. We further introduce entropy-weighted, angle-aware, and density-aware OT variants to capture uncertainty, directional drift, and sparse-support behavior. Evaluation on DARPA Transparent Computing data spanning Linux, Windows, BSD, and Android, across two APT scenarios and twelve cross-OS transfer pairs, shows that the proposed framework improves ROC-AUC and nDCG over source-only anomaly-detection baselines. The results demonstrate that source-only provenance modeling, combined with semantic abstraction and OT-based anomaly scoring, can support practical cross-platform APT detection without target-domain supervision.

Sidahmed Benabderrahmanea, Petko Valtchev, James Cheney +1

New York University, NYUAD, Division of Science, Computer Science Department. · University of Quebec in Montreal, Computer Science Department, Montreal. · University of Edinburgh, School of Informatics, Edinburgh.

Cross-Domain Transfer Learning Optimal Transport Anomaly Detection

Jun 7, 2026

RadOT-Eval: Auditable Structured-Evidence Transport for Radiology Report Evaluation

Automatic evaluation is critical for high-stakes text generation, where errors often involve omitted findings, hallucinated content, polarity reversals, location changes, uncertainty mismatches, and temporal-comparison errors rather than low surface similarity alone. Radiology report generation provides a challenging test case because generated reports must preserve structured clinical evidence across sources. We present RadOT-Eval, an interpretable structured-evidence optimal transport framework for offline auditing of radiology report generation. RadOT-Eval decomposes reference and candidate reports into attribute-structured clinical evidence units, aligns corresponding evidence using entropy-regularized optimal transport, and uses clinically meaningful side-channel discrepancies in a monotone risk model to predict error burden. All transport, feature, and readout choices are selected using the ReXVal dataset, and the frozen system is evaluated on the independent RadEvalX dataset. RadOT-Eval achieves Spearman correlations of 0.715, 0.548, and 0.399 with total, clinically significant, and clinically insignificant annotated error burden, respectively, yielding higher point estimates than standard evaluation metrics and the open-source large language model (LLM)-based evaluator GREEN-radllama2-7B. In a frozen auxiliary corruption-sensitivity stress test on ReXErr-v1, RadOT-Eval achieves 0.768 AUROC and a 0.990 corrupted-greater-than-clean paired win rate. These results show that structured evidence transport provides an auditable, rank-oriented evaluation tool for high-stakes generated clinical text under ReXVal-only model selection and frozen RadEvalX testing.

Weixin Liu, Juming Xiong, Yang Li +5

Vanderbilt University, Nashville, TN, USA · Vanderbilt University Medical Center, Nashville, TN, USA · Virginia Tech, Blacksburg, VA, USA

Radiology Report Generation Automated Evaluation Optimal Transport Radiology Report Evaluation

Jun 6, 2026

MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport

Understanding and comparing structures in scalar fields is a central challenge in scientific visualization, with applications ranging from feature analysis to temporal and structural comparison. The Morse-Smale (MS) complex provides a natural representation by decomposing a scalar field into regions induced by gradient flow. However, existing approaches typically rely on graph-based representations, matching critical points without explicitly establishing correspondences between regions. In this work, we represent the MS complex as a hypergraph, where critical points form nodes and regions define hyperedges. We introduce MS-COOT, a co-optimal transport distance that jointly computes correspondences between critical points and regions. This formulation enables explicit region-to-region matching within a distance-based framework, allowing identification of region-level events such as splitting and merging. We instantiate this framework with domain-specific components, including a hypernetwork function encoding relationships between critical points and regions, persistence-based probability measures, and a sample cost term that incorporates critical point attributes. We evaluate MS-COOT on datasets spanning 2D simulations, 3D surface meshes, and volumetric data. Our results show that MS-COOT provides distinct characterization of certain region-level structural changes compared to graph-based distances, achieves competitive accuracy in shape classification, and improves classification of simulation parameter settings in a 3D ensemble.

Guangyu Meng, Mingzhe Li, Erin Wolf Chambers

Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, Indiana, United States

Optimal Transport Topological Data Analysis

Jun 6, 2026

Conditional Random Ordered Transport Spaces

A small Wasserstein distance does not certify that a transformation is admissible. In evidence-constrained, semantic, causal, physical, monotone, or risk-sensitive learning, one must ask not only how far two probability laws are, but whether mass has moved in a direction allowed by available information. We introduce conditional random ordered transport spaces (CROTS), a class of

L^0

-valued spaces of random probability measures equipped with a Wasserstein ambient metric, a closed stochastic order, hard and soft ordered transport discrepancies, and a conditional risk functional for evaluating order violation under an evidence sigma-field. The central object is an order-admissible transport geometry for random measure-valued dynamics, distinct from cone-valued metrics, ordered Kantorovich constructions, random Wasserstein spaces alone, and model-specific residuals for generative paths. We develop the foundations of CROTS as a space theory for reliable distributional learning. The results include well-posedness and duality for hard and soft ordered transport, soft-to-hard variational convergence, measurability and completeness of the random lifted space, reductions to classical Wasserstein and ordered geometries, ordered geodesics, constrained barycenters and projections, conditional risk-transport duality, and separation of order-violating distributions. The main stability theorem shows that random learning dynamics may converge in the ambient Wasserstein metric while its local admissibility leakage follows a separate conditional order-risk recursion. The resulting asymptotic order-risk floor provides a mathematical language for evidence overreach, ordered distribution shift, robustness failure, and admissible distributional dynamics.

Lei Luo, Jian Yang

PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education School of Computer Science and Engineering Nanjing University of Science and Technology Nanjing, China · PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education2026 School of Computer Science and Engineering Nanjing University of Science and Technology Nanjing, ChinaJun

Optimal Transport

Jun 6, 2026

Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps. In Euclidean space, barycentric projection converts a coupling into a map by taking conditional expectations, but on a Riemannian manifold curvature and cut loci make this operation nontrivial. We develop a framework for barycentric projections of transport couplings on Riemannian manifolds. The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss. The corresponding minimum value is an integrated conditional Fréchet variance, which vanishes exactly for map-induced couplings and therefore defines a conditional-variance Monge defect. We also study a tangential log-exp projection, prove its Euclidean exactness, its compatibility with Brenier-McCann maps in the Monge case, and its interpretation as the first unit Riemannian gradient update for the intrinsic objective. For discrete couplings, both constructions decompose row-wise into weighted Fréchet mean and log-exp problems. Experiments on spherical data, synthetic SPD data, and real EEG covariance matrices support the proposed division of roles: the intrinsic projection is the variational representative, while the tangential projection is a useful local displacement surrogate.

Kisung You

Department of Mathematics Baruch College

Riemannian Geometry Optimal Transport

Jun 5, 2026

A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space

We study the minimax rate of estimating a future value

μ_{t_n+h}

of a curve

t\mapstoμ_t

in the

2

-Wasserstein space

\mathcal{P}_2(\mathbb{R}^d)

from finitely many noisy snapshots of its past, under an adiabatic bound

\|\nabla_t^k v\|\le\varepsilon

on the

k

-th covariant derivative of the velocity field. Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every estimator incurs

W_2

-risk with

M

-exponent

γ_d(k+1)/(k+1+γ_d)

,

γ_d=\min(1/d,1/2)

(

M

the total sample size). It follows from a temporal-to-spatial reduction: the smoothness budget defines a reachable

W_2

-ball into which a transport packing is embedded along the time axis, and the information of the entire snapshot experiment is controlled by a Fano argument -- the spatial packing is classical, but its smoothness-admissible temporal embedding and the full-window analysis are new. The bound interpolates a dimension-free extrapolation floor of order

\varepsilon h^{k+1}

-- the irreducible cost of an unobserved future, present even with the exact past -- and the spatial estimation curse

M^{-γ_d}

, recovering the static distribution-estimation rate as

k\to\infty

. We state the lower bound in a design-dependent form -- with a design-weighted effective sample size -- valid for arbitrary observation times, and obtain the closed-form exponent in the dense (equispaced) regime. The matching upper bound is established at

k=0

(rate

M^{-1/(d+1)}

,

d\ge3

) and, in a translation submodel, for all

k

; for

k\ge1

a covariant estimator attains the rate conditionally on two estimates (a comparison-geometry bias bound and an optimal-transport map-estimation rate), leaving the unconditional general-

k

upper bound as an open problem. Numerical experiments on synthetic curved and flat families corroborate the predicted exponents.

Munsik Kim

Minimax Estimation Spatiotemporal Forecasting Optimal Transport Wasserstein Distance

Jun 4, 2026

Your GFlowNet Secretly Learns an Optimal Transport Plan

Generative Flow Networks (GFlowNets) are a framework for sampling structured objects via stochastic trajectories in a directed graph. In this work, we establish a theoretical connection between non-acyclic GFlowNets and optimal transport (OT). We show that fixing the initial flow distribution in a minimum-flow GFlowNet reduces its objective to a Kantorovich OT problem with graph-induced shortest path costs. At the optimum, the learned GFlowNet policy therefore encodes an optimal transport plan from the source distribution to the target distribution: we show that sampling trajectories from the minimum-flow GFlowNet recovers the corresponding optimal coupling. Our formulation enables applying the GFlowNet learning framework to OT problems on large graphs via edge flows and neural parameterization. Experiments confirm agreement with exact OT solvers and demonstrate that GFlowNets can learn high-quality transport plans.

Ian Maksimov, Nikita Morozov, Denis Belomestny +1

1HSE University · 2Duisburg-Essen University

Generative Flow Networks Flow-Based Generative Modeling Optimal Transport

Jun 3, 2026

Multimarginal flow matching with optimal transport potentials

Flow matching (FM) has emerged as a powerful framework for learning dynamic transport maps between two empirical distributions. However, less explored is the setting with intermediate observed marginals that can help constrain the flows between the endpoints. This "multimarginal" regime is central to modeling temporal evolution in dynamical systems in many scientific domains that can sample sequential distributions. We tackle this problem with a novel approach that leverages the connection between FM and dynamic optimal transport (OT), softly steering the flow towards the intermediate marginals through potential terms in the dynamic OT action. By extending the conditional FM learning target to incorporate these potentials, we derive an efficient, simulation-free algorithm for multimarginal FM that offers considerable flexibility in the spatiotemporal dynamics of the learned flows. We demonstrate state-of-the-art performance and training efficiency of OT-potential FM (OTP-FM) on diverse single-cell RNA sequencing, oceanographic, and meteorological datasets. Our code is available at https://github.com/Bexorg-Inc/OTP-FM.

Raghav Kansal, David Crair, Nghia Nguyen +2

Bexorg, Inc., New Haven, CT, USA.

Flow Matching Flow Map Learning Optimal Transport

Jun 3, 2026

Cone-Compatible Monge Geometry for High-Dimensional Ordered Optimal Transport

High-dimensional optimal transport is seldom available in closed form. The one-dimensional case is exceptional because the order of the real line is compatible with convex transport costs, making monotone rearrangement optimal. This paper studies when an analogous Monge structure can be recovered in higher dimensions from a partial order. We introduce a cone-compatible Monge geometry: a closed convex cone (K) induces the order (x\preceq_K y) whenever (y-x\in K), and is compatible with a cost if ordered pairs satisfy a Monge exchange inequality. For squared Mahalanobis costs (c_M(x,y)=(x-y)^\top M(x-y)), we prove a sharp characterization: compatibility holds exactly when (K) is acute under the (M)-inner product, namely (u^\top Mv\ge0) for all (u,v\in K), equivalently (K\subseteq K_M^*). Under this condition, measures supported on cone chains admit a quantile-type closed-form optimal coupling, yielding exact transport under the original ground cost rather than after projection or metric replacement. We distinguish the resulting cone-chain Wasserstein metric on canonically ordered chain distributions from an extended directed cone transport cost on general measures, and develop feasibility, duality, stability, approximation, Gaussian recovery, statistical, and computational results. The theory is complementary to sliced and tree Wasserstein distances: it is not a universal fast surrogate, but a way to obtain interpretable, direction-valid, original-space monotone transport for ordered high-dimensional data.

Lei Luo, Hongliang Zhang, Jian Yang

PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education, School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China

Optimal Transport Wasserstein Distance

Jun 2, 2026

Optimal Transport Flow Matching by Design

Flow matching models learn to transport samples from a simple prior distribution to a complex data distribution. When prior-data pairs are coupled via optimal transport (OT), the learned trajectories are straight and non-crossing, enabling fast, even single-step, generation. However, computing the OT coupling in high dimensions is intractable, and existing methods attempt to solve the OT problem, at the cost of persistent bias or significant overhead. Rather than solving for the OT coupling, we reformulate the problem. Once the prior is treated as a design choice rather than a fixed input, the OT coupling between prior and data is no longer unique. Many priors admit an OT-optimal identity coupling to the data, leaving us free to choose one that is also tractable to sample. We identify low-frequency projection of natural images as such a choice. The identity coupling between data and its low-frequency representation is empirically OT-optimal, the prior is structured enough to be sampled by a lightweight model at inference, and the remaining flow-matching task reduces to synthesizing high-frequency detail. Interpolating the prior with Gaussian noise further improves generation quality while preserving the OT coupling. The approach requires no modifications to the flow model itself, and integrates naturally with latent-space models, classifier-free guidance, and one-step generation frameworks. Across all benchmarks, our method reduces trajectory curvature by more than

2\times

compared to existing flow matching methods, yielding better generation quality in the few-step regime.

Shimon Malnick, Matan Rusanovsky, Ohad Fried +1

1Tel Aviv University · 2Reichman University

Flow Matching One-Step Generative Modeling Flow-Based Generative Modeling Optimal Transport

Jun 1, 2026

A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solution

Optimal transport (OT) provides a principled framework for mapping between probability distributions. Despite extensive progress, applying OT to large-scale data remains computationally demanding, and the resulting pointwise transport plans are often difficult to interpret. We introduce Optimal Mixture Transport (OMT), a scalable framework that shifts the transport paradigm from individual samples to mixtures of subpopulations, reformulating the transport problem as a strictly biconvex optimization with a unique global minimizer. We further establish theoretical guarantees on the stability of the OMT map, showing that bounded perturbations of the underlying distributions lead to bounded changes in the transport plan. By formulating subpopulations as exponential-family distributions, OMT decouples computational complexity from the sample size, scaling solely with the number of mixture components. We demonstrate the effectiveness and practicality of OMT on a wide range of synthetic benchmarks and real-world datasets, including image data and large-scale single-cell RNA sequencing measurements.

Yeganeh Marghi, Kelly Jin, Uygar Sümbül

Allen Institute Seattle, WA, USA

Optimal Transport

Jun 1, 2026

Network Learning with Semi-relaxed Gromov-Wasserstein

Estimating the generative mechanism of large-scale networks is a fundamental challenge in statistical machine learning. It requires the identification of the latent connectivity structure, which is in general an NP-hard combinatorial problem due to the absence of canonical node labels. We address this challenge by allowing for probabilistic couplings, thereby relaxing the assignment problem. Our estimation framework can be formulated as a semi-relaxed Gromov-Wasserstein objective and provides a low-dimensional representation of the generative structure. We solve this via a block-coordinate conditional gradient algorithm. Despite the relaxation, the resulting solution is typically deterministic: in fact, we show that the optimality gap between the relaxed solution and the deterministic assignment vanishes at rate

O(1/n)

, where

n

is the number of nodes. This allows for tractable recovery of the underlying model and enables rigorous statistical analysis: we establish consistency and minimax-optimal convergence rates for both stochastic block models and Holder-smooth graphons. Our implementation scales efficiently with

n

, as demonstrated on both synthetic and real-world datasets.

Charles Dufour, Ulysse Naepels, Leonardo V. Santoro

EPFL, Institute of Mathematics Lausanne, Switzerland

Graph Structure Learning Latent Variable Models Stochastic Block Model Statistical Learning Theory Optimal Transport

Jun 1, 2026

Convex Distance Operator Transport: A Convex and Geometry-Preserving Formulation

We introduce Convex Distance Operator Transport (CDOT), the first convex optimal transport framework that aligns distributions across heterogeneous domains by jointly preserving feature correspondence and intrinsic geometric structure. Specifically, CDOT employs an operator-based regularization that aligns aggregated distance structures by introducing distance and conditional expectation operators. Consequently, the proposed regularization improves the robustness to local geometric variations. We further prove that the resulting CDOT discrepancy is a valid pseudometric on the space of attributed compact metric-measure spaces. In addition, we characterize the relationship between CDOT and Gromov--Wasserstein (GW) through a new notion of dispersion gap, formally elucidating the geometric source of non-convexity in GW compared to the convexity of CDOT. In the finite-sample regime, we derive a non-asymptotic risk bound decomposed into optimization and statistical errors, establishing risk consistency under a globally convergent Frank--Wolfe algorithm. Experiments on synthetic point clouds, brain connectomes, and graph classification benchmarks demonstrate better performance over existing methods, with stable and reliable behavior in practice.

Junhyoung Chung, Euijong Song, Won Hwa Kim +1

KRAFTON · Department of Statistics, Seoul National University, Seoul, South Korea · Graduate School of AI, POSTECH, Pohang, South Korea +3

Optimal Transport

May 30, 2026

Generate in Reconstruction Space, Match in Semantic Space: Transport Geometry for One-Step Generation

Generative modeling and self-supervised representation learning (SSL) optimize structurally different objectives: generative training rewards distributional fidelity, while SSL rewards semantic coherence. Yet recent work repeatedly finds that SSL features improve generative training, though the mechanism of this synergy remains unclear. Here, we study the benefits of SSL in generative modeling in the framework of one-step generation where the role of representation is explicit: frozen SSL features are used to match generated samples to real data. We use the Sinkhorn divergence in that feature space, providing a tractable surrogate for the Wasserstein distance, the population-level discrepancy approximated by Fréchet-style evaluation metrics (such as FID). We find that this objective becomes highly effective when computed in a semantically structured SSL feature space (a 39

\times

reduction in ImageNet FID). We trace this behavior primarily to matching estimation: semantic SSL features that suppress nuisance reconstruction details induce a more compact geometry, making distribution matching more tractable. As a consequence, the best training SSL features need not match the features used by the evaluation metric. In particular, we show that using Inception as the feature extractor can improve FID while degrading matching stability and sample quality, revealing a form of metric hacking. Using extensive experiments on ImageNet, we identify which SSL feature families lead to best generation performance and show that matching stability is a quantitative criterion for selecting them. Code is available at https://github.com/Genentech/semantic-transport-generation.

Hugues Van Assel, Edward De Brouwer, Saeed Saremi +2

Genentech

One-Step Generative Modeling Distribution Matching Self-Supervised Visual Representation Learning Optimal Transport

May 28, 2026

CellBRIDGE: Learning Cellular Trajectories via Interaction-Aware Alignment

Inferring dynamics from population snapshots is a fundamental challenge in machine learning and biology. In scRNA-sequencing (scRNA-seq), destructive measurements preclude direct tracking of individual cells across time, making trajectory inference underdetermined. Optimal Transport (OT) provides a principled framework for snapshot alignment, but a long-standing modeling question is which cost functions yield biologically meaningful couplings. Standard OT approaches rely on gene-expression distances, implicitly treating cells as independent points and neglecting structured cell-cell communication mediated by ligand-receptor signaling. We introduce CellBRIDGE (Cell-Based Regularized Interaction-Driven Gene Expression), which augments feature-based OT with a directed, typed interaction cost derived from ligand-receptor activity. By explicitly modeling cell-cell communication, CellBRIDGE improves cross-snapshot couplings and downstream trajectory estimates across synthetic and real scRNA-seq datasets relative to feature-only baselines. Notably, CellBRIDGE enables mechanistically interpretable in silico perturbations: on lung cancer data, silencing specific ligand-receptor pairs induces trajectory shifts that recapitulate expected effects of targeted pathway inhibition.

Silas Ruhrberg Estévez, Nicolas Huynh, Tennison Liu +4

DAMTP, University of Cambridge · Francis Crick Institute · University of Colorado Anschutz Medical Campus

Single-Cell Transcriptomics Optimal Transport Single-Cell RNA Sequencing

May 27, 2026

Semantic Optimal Transport for Sparse Autoencoder Feature Matching and Circuit Compression

Sparse autoencoders (SAEs) have become a central tool for interpreting language models. However, two key SAE analyses that remain difficult to scale are (1) matching semantically similar features across multi-layers and (2) compressing large feature circuits into interpretable supernodes. Although these have been treated as separate problems, we show that both are instances of a more fundamental challenge, which we frame as the estimation of semantic distances between SAE features that lie on different activation manifolds. We introduce a distributional framework for this problem, in which each feature is represented not by a single decoder vector like in the literature, but by an activation-weighted distribution over the hidden states that express it. By projecting these distributions into a shared reference space and comparing them with Wasserstein distance, our method provides a unified semantic metric for cross-layer feature comparison. We prove that our representation is invariant to activation rescaling, stable under perturbations, and recovers true matches under finite-sample margin conditions. Empirically, our method outperforms decoder-vector and LLM-based baselines and captures subtle functional distinctions between related features. Notably, our method compresses large feature circuits into interpretable supernodes automatically.

Tue M. Cao, Nguyen Do, My T. Thai

Department of Computer and Information Science and Engineering, University of Florida

Sparse Autoencoders Mechanistic Interpretability Neural Network Compression Optimal Transport

May 27, 2026

Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models

We study inference in stochastic block models (SBMs) through the lens of optimal transport (OT). We first establish that maximum likelihood variational inference (MLVI) can be interpreted as a semi-relaxed Gromov-Wasserstein (srGW) projection with entropic regularization. While this formulation yields accurate clustering, the entropic regularization prevents transport plans to be sparse, hindering intrinsic model selection. Consequently, we investigate unregularized srGW estimators, and prove that they consistently recover both the SBM connectivity matrix and latent cluster assignments in the asymptotic regime. However, this asymptotic property does not translate into reliable model selection in finite samples, and calls for additional mechanisms to promote sparsity in the inferred cluster proportions. We empirically show that such a regularized formulation yields estimators that simultaneously recover model parameters and select the number of clusters in a single optimization problem, thereby avoiding costly grid search or heuristic model selection procedures.

Simon Queric, Cédric Vincent-Cuaz, Charles Bouveyron +1

Universit´e Cˆote d’Azur, Inria, CNRS LJAD, Maasai Nice, France · EPFL Lausanne, Switzerland · Universit´e Cˆote d’Azur, Inria, CNRS CEPAM, Maasai Nice, France

Model Selection Latent Variable Models Stochastic Block Model Graph Clustering Variational Inference Optimal Transport

May 26, 2026

MST-Direct at Scale: Multivariate and Conditional Geostatistical Simulation via Sinkhorn Optimal Transport

This paper extends MST-Direct, a Matching-via-Sinkhorn-Transport approach for multivariate geostatistical simulation, from the original bivariate, unconditional, small-grid formulation to multivariate, conditional, and large-grid settings. We address the three main limitations identified in the original work: (i) scalability beyond a few thousand nodes through a sparse, candidate-restricted Sinkhorn matcher with O(nC) memory complexity; (ii) extension to multiple variables by matching target value tuples onto an independent FFT-MA Gaussian backbone that reproduces a prescribed variogram; and (iii) hard-data conditioning by fixing observed data tuples at their spatial locations while conditioning the backbone through kriging. Because the transport plan remains a permutation of the target tuples, the multivariate joint distribution is preserved exactly. The method is validated using the same six-variate, heteroscedastic, strongly nonlinear reference distribution employed in Direct Multivariate Simulation (DMS), under both unconditional (200x200) and conditional (100x100, 200 hard-data samples) scenarios, and is benchmarked against the Projection Pursuit Multivariate Transform (PPMT). Results show that MST-Direct reproduces the joint distribution with zero histogram error, exactly honours hard data, and accurately reproduces the prescribed spatial correlation structure, whereas PPMT remains an approximation. Index Terms-Optimal transport, Sinkhorn algorithm, geostatistical simulation, multivariate simulation.

Tcharlies Bachmann Schmitz

Data Science, PX.Center Joinville, Brasil

Optimal Transport

May 26, 2026

Supervised Distributional Reduction via Optimal Transport and Dependence Maximization

Learning representations that capture both intrinsic data geometry and target-relevant structure remains a fundamental challenge, particularly in settings where data reduction must balance compression with predictive fidelity. While distributional reduction-encompassing joint clustering and dimensionality reduction-offers a principled way to summarize data, its supervised variants remain relatively under-explored, despite the importance of retaining task-relevant signal for downstream prediction and decision-making. We propose Supervised Distributional Reduction (SDR), an algorithm for learning target-aware representations by combining optimal transport with explicit dependence maximization. SDR builds on the Fused Gromov-Wasserstein (FGW) objective to align the relational structure of the input distribution with a set of representative points, while augmenting it with a direct dependence term that encourages the learned embeddings to capture predictive signal more explicitly. This results in compact representations that reflect both geometric structure and supervision. Beyond representation learning, SDR naturally induces a data-dependent, non-stationary geometry that can be leveraged for settings such as Gaussian Process (GP) modelling. By redefining distances through target-aware distributional alignment, SDR enables the construction of adaptive kernels that respond to local variations in both data geometry and supervision, offering an optimal transport-based perspective on non-stationary kernel design.

Sai-Aakash Ramesh, Archit Sood, Andrew Corbett +1

digiLab, UK · University of Bristol, UK

Kernel Methods Dimensionality Reduction Optimal Transport

May 25, 2026

Learning Permutation from Structure Without Supervision

Many learning problems require uncovering a hidden ordering that reveals structure in unordered data, such as monotonicity in sorting or spatial continuity in jigsaw reconstruction. In these settings, permutations can be learned as latent operators by optimizing objectives defined directly on the reordered output, often without access to ground-truth orderings. Differentiable relaxations such as Gumbel-Sinkhorn make this approach practical by approximating permutation matrices with doubly stochastic matrices. However, learning from structure without supervision induces a non-uniform uncertainty: some assignments become confident early, while others remain ambiguous. Existing methods control this process using a single global temperature, forcing all assignments to sharpen or diffuse simultaneously and leading to instability at scale. We introduce an entropy-adaptive formulation of Gumbel-Sinkhorn that locally modulates temperature based on assignment uncertainty. This allows confident assignments to discretize early while preserving exploration where uncertainty remains. Across sorting and jigsaw reconstruction tasks and in routing-style settings, adaptive entropy control improves training stability and final permutation quality relative to fixed-temperature baselines, particularly as problem size and assignment ambiguity increase.

Ran Eisenberg, Ofir Lindenbaum

Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel. · Faculty of Engineering, Bar-Ilan

Unsupervised Learning Jigsaw Puzzle Solving Optimal Transport

May 24, 2026

Lifted Schrödinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions

We study stochastic density control between Gaussian-mixture endpoint distributions under Brownian prior dynamics. Since the direct Schrödinger bridge between Gaussian mixtures is generally not available in closed form, we introduce a lifted path-space construction in which each trajectory is augmented with a source--target component label. Consequently, the problem decomposes into Gaussian component-to-component Schrödinger bridges with explicit marginal, drift, and cost formulas, while the mixture-level assignment reduces to a finite-dimensional entropic coupling problem with a Sinkhorn scaling form. We then analyze the projection obtained by discarding or forgetting the label. By construction, the projected law satisfies the original Gaussian-mixture endpoint constraints, but its relative entropy generally differs from the lifted relative entropy by a nonnegative conditional label-information gap. This gap reveals a path-space obstruction: the lifted optimizer cannot, in general, be identified with the direct unlabeled Schrödinger bridge after projection. We also derive the posterior-averaged Markov drift associated with the projected marginal flow, prove a kinetic-energy upper bound, and identify a common path-potential condition under which the projection gap vanishes. Several numerical illustrations showing density and shape control are recorded for a self-contained exposition.

Siddhartha Ganguly, George Rapakoulias, Panagiotis Tsiotras

S. Ganguly, G. Rapakoulias, and P. Tsiotras are with Daniel Guggenheim School of Aerospace, Georgia Institute of Technology, ½ Atlanta, USA.

Stochastic Optimal Control Schrödinger Bridges Optimal Transport

May 23, 2026

AvAtar: Learning to Align via Active Optimal Transport

Alignment plays a fundamental role in many machine learning problems, such as multi-network analysis, multimodal learning, and point cloud registration. Recent works increasingly leverage optimal transport (OT) for distributional alignment, whose effectiveness largely depends on sparse supervision that is hard or costly to obtain in practice. Existing works, however, largely overlook how to actively acquire high-quality supervision to improve their alignment performance under OT frameworks. In this paper, we propose a principled active alignment framework for optimal transport alignment called AvAtar. We quantify the informativeness of a candidate by measuring its gradient-based impact on the global alignment result, computed as the gradient propagation from the global alignment result to all possible supervisions of the candidate through the entropy-regularized OT formulation. While differentiating through OT is challenging given its constrained nature, we leverage the adjoint-state method to reformulate the computation to a linear system solvable by the conjugate gradient method with linear complexity and guaranteed convergence. By encoding the global alignment result via effective utility functions, AvAtar is applicable to general alignment problems under the OT framework. Extensive experiments on three representative alignment tasks demonstrate the effectiveness, scalability, and generalizability of the proposed AvAtar.

Qi Yu, Ruizhong Qiu, Zhichen Zeng +3

University of Illinois Urbana-Champaign · University of Florida · 3Arizona State University

Active Learning Optimal Transport