Optimal Transport

We study statistical rates in entropic optimal transport in the semi-discrete regime where one measure has finite support and the other is subGaussian. Our main result establishes parametric convergence rates for the empirical dual potentials to their population counterparts, with no dimension dependence in the leading term. Our result relies on tailored strong concavity analysis of the semi-dual objective, coupled with specialized bounds for the semi-discrete potentials. As a consequence, we obtain fast rates for downstream quantities derived from the optimal coupling. Chiefly, the empirical barycentric projection achieves a squared-error rate $n^{-1}$, matching the fully compact case and improving over the less favorable $n^{-1/2}$ rate known for fully subGaussian settings. Altogether, these results may indicate a lower complexity adaptation phenomenon whereby the statistical complexity of the barycentric projection is governed by the discrete measure. As an application, we analyze Sinkhorn-EM, an EM-type algorithm in which the E-step is replaced by an entropic optimal transport problem. In a well-specified and balanced two-component Gaussian mixture model, we prove $\sqrt{n}$-consistency of the empirical iterates to their population counterparts for any fixed number of iterations, matching classical EM rates up to a $\sqrt{\log n}$ factor. Simulations support the theory.

Graph prompt learning enables parameter-efficient adaptation of frozen Graph Neural Networks to downstream tasks through lightweight prompt parameters. As routing becomes increasingly node-adaptive, however, independently optimized local decisions can collectively concentrate assignment mass on a small subset of a finite shared prompt bank, even when individual node--prompt matches remain locally meaningful. We propose MINT (Measure-INtegrity Transport), an entropically regularized optimal transport framework that formulates node-to-prompt adaptation as a globally coupled allocation problem. The transport cost favors local geometric compatibility, while a prescribed prompt-side marginal explicitly controls graph-wide prompt utilization. We further derive an exact variance decomposition that separates prompt-side geometric variance into retained prompt-update variation and within-node barycentric dispersion, together with a conditional stability bound for the frozen-encoder forward map. Across standard citation networks and additional heterophilic graphs, MINT remains competitive in few-shot adaptation. Controlled and end-to-end experiments further distinguish the roles of routing and topology: fixed-marginal routing controls graph-wide prompt utilization and has measurable end-to-end effects on citation networks, while topology augmentation provides a complementary, graph-dependent mechanism for addressing structural mismatch. Code is available at https://github.com/Ga1axy0051/MINT.

Gromov--Wasserstein optimal transport (GW-OT) extends classical optimal transport by introducing structure-aware transport cost. This is particularly relevant for spatial transcriptomics, where dynamical reconstruction should preserve tissue structure in addition to matching expression patterns. While static formulations have been widely used for such structure-aware alignment, a general dynamic formulation for reconstructing continuous trajectories is still missing. We introduce Travelling Pair Dynamical Alignment and Trajectory Estimation (TP-DATE), a theoretical and computational framework to generalize GW-OT dynamically in a simulation-free manner. We formulate a broad class of static and dynamic Quadratic-form OT (QOT) through path actions and prove the static dynamic equivalence. We further develop travelling-pair flow matching, which allows interacting conditional paths and marginalizes their interactions into a single vector field. On synthetic and real spatial transcriptomics data, TP-DATE better preserves spatial structure and improves continuous 3D dynamics reconstruction.

Probabilistic certification of Bayesian neural networks lower-bounds the posterior probability that a model satisfies a verifier-defined safety property. In one-shot federated Bayesian learning, however, the deployed model is obtained by aggregating parameters drawn from client-specific posterior distributions, so local certificates do not directly guarantee safety of the aggregated model. This paper develops a deployment-consistent certification framework by propagating local posterior events through the deployment aggregation rule, with an exact geometric characterization for Federated Averaging (FedAvg). Each client constructs disjoint hyper-rectangular regions in parameter space and computes their probability masses. The server forms Cartesian products of these regions, maps them through the deployment rule, and retains a product event only when its aggregation image is verified to satisfy the safety property. Under independent client posteriors, each product-event probability factorizes into local masses, and summing verified disjoint events yields a lower bound on safety probability of the deployed model. For FedAvg with nonnegative aggregation coefficients, the image of a Cartesian product of axis-aligned hyper-rectangles is exactly a weighted hyper-rectangle, introducing no set over-approximation. We distinguish the proposed transported-event certificate from direct certification under posterior distributions induced by FedAvg and Product-of-Gaussians aggregation. Experiments on MNIST and Fashion-MNIST under label-Dirichlet heterogeneity show that the transported FedAvg certificate ranges from 22.51% to 46.89%, while direct global certificates range from 72.05% to 91.39%. Results show that predictive accuracy and certifiable safety do not necessarily follow the same trend, and that global posterior constructions can exhibit distinct certification behavior across architectures.

This letter presents a unified formulation and a controlled numerical comparison of generative-model approaches to the nonlinear filtering problem. Under this formulation the analysis step is realized by a transport of the forecast distribution to the posterior, the approaches differing only in how that transport is selected and learned. We derive three new filters, based on stochastic interpolants, their deterministic flow-matching limit, and Schrödinger bridges realized through forward--backward SDEs. We develop a two-stage tuning procedure that separates the training of the generative model from its online refinement. The resulting methods are compared against the optimal transport filter (OTF), the Knothe--Rosenblatt filter (KRF), the sequential importance resampling (SIR) particle filter and the ensemble Kalman filter (EnKF), in terms of accuracy, computational time, and sensitivity to ensemble size and state dimension. The results indicate that every generative filter resolves multimodal posteriors that the EnKF and SIR do not, that no single generative framework dominates, the preferred method being set by the available online budget and ensemble size, and that the filters differ in the regularity of the particle trajectories they produce.

The last decade has seen the development of powerful methods for learning complex structure from high-dimensional data. These advances have brought to the foreground fundamental connections between subdisciplines of physics, applied mathematics, and machine learning. In this review, we bring together some of these ideas, often expressed in different languages, to highlight a conceptual thread that links five distinct fields: control theory, optimal transport, probabilistic inference, non-equilibrium thermodynamics, and machine learning. A common theme is the optimization of free-energy-like functionals under dynamical or statistical constraints. We offer a guided tour through this thread and present selected applications in reinforcement learning, variational inference, and generative modeling. The review does not assume prior familiarity with these topics, and begins with principles originating from physics.

Offline reinforcement learning aims to learn a policy solely from fixed datasets, which often contain multimodal action distributions. Flow policies can naturally represent such multimodal behaviors, but learning an efficient one-step flow policy remains challenging: standard value guidance often leads to mode collapse or exploits overestimation bias in out-of-distribution regions. To address this, we introduce One-step Flow policy via Optimal Transport (OptiFlow), a framework for one-step flow policy learning as a structured sample-allocation problem. OptiFlow jointly trains a value-aware reference flow policy and an efficient one-step policy, coupling their action samples through state-wise entropic optimal transport. For each state, critic-estimated values define the priority of distillation target actions, while the action-distance cost ensures geometrically compatible pairings. By avoiding direct critic maximization, our transport-guided approach enables in-distribution exploitation by anchoring the one-step policy to high-value, dataset-supported modes without the risk of out-of-distribution divergence. Experimental results demonstrate that OptiFlow effectively captures optimal multimodal behaviors and achieves strong performance across diverse offline RL benchmarks. Our code is available at https://github.com/Yonsei-DILLab/OptiFlow.

Learning a motion prior requires a reward that guides a policy from its current behavior toward demonstrated motion. Adversarial Motion Priors (AMP) provide such a reward with a discriminator. However, adversarial objectives can become uninformative when policy and expert supports are far apart. A naive use of optimal transport (OT) averages matched expert successors into a barycentric target. Averaging across gait phases can weaken the target's joint motion. We introduce Flow-Matched Motion Priors (FMP), an online scalar reward learned from paths connecting current rollout histories to an expert motion bank. Entropic OT supplies the coupling. Before each policy update, we train a neural potential with flow matching (FM) along the rollout-to-expert paths, endpoint-gradient supervision, and relative-value calibration. The actor receives only physical observations and the reward remains a scalar, as in AMP. Controlled reward-model experiments show substantially better generalization beyond the fitting rollout than value-only or endpoint-only fitting. On Unitree G1, matched 50-million-transition experiments compare FMP with AMP, a barycentric OT reward, and nested ablations under demonstration and fixed-pose initialization. FMP produces stable forward walking at 0.727 m/s from demonstration resets and 0.338 m/s from a fixed default pose. In the fixed-pose condition, it incurs 129 falls versus 243 for the endpoint-only control. Against a static score-gradient teacher, dynamic FM reduces score-increment error at interpolation fractions 0.25 and 0.50 while using 29% less offline fitting time.

End-to-end Supervised Graph Prediction (SGP) requires a permutation-invariant loss to compare predicted and target graphs with arbitrary node orderings. Such losses typically involve a costly graph-matching problem. We first study three Optimal Transport relaxations of this problem and show, theoretically and empirically, that the Gromov-Wasserstein (GW) objective is the most suitable for SGP. Then, to avoid solving the resulting inner optimization for every training example, we propose to amortize the graph matching (node alignment) problem. For each training sample, the loss function leverages a transport plan provided by a parametric matcher based on the differentiable Sinkhorn algorithm applied on empirical node distributions. The graph prediction module and the matcher are jointly learned. We showcase the efficiency of this approach on toy and real world SGP problems of increasing complexity including a novel Mass-spectra to Scaffold task that we introduce.

Methods of Branched Optimal Transport (BOT) mimic the economy and efficiency of natural tree-like structures, such as those found in rivers and biological systems. These methods are widely applicable for designing efficient networks in society, from river basins and blood vessels to mail and gas distribution systems. However, they remain understudied in the context of designing deep generative models, particularly at a large scale. Standard continuous-time generative models, such as the flow matching approach, fail to capture the inherent hierarchical and branching patterns present in real-world data. Current models provide no mechanism for flows to merge or share pathways to minimize total transport cost. Inspired by the "economy of scale" principle in BOT, we introduce a novel, scalable branched flow-matching algorithm designed to solve the branched optimal transport problem in high dimensions. Our method adapts the Benamou-Brenier continuous-time optimal transport formulation to learn branched generative flows. These flows allow probability mass to aggregate along common pathways before branching out to diverse targets. Parametrized by neural networks, our method effectively learns complex branched generative processes. We demonstrate its effectiveness on challenging high-dimensional tasks in biology and image generation.

Optimal transport (OT) compares distributions and aligns datasets in machine learning, yet unregularized discrete OT requires a linear program with quadratically many transport variables. We propose HELLO, a hierarchical solver that casts large-scale discrete OT as edge localization and uses dual potentials to guide both coarse-to-fine initialization and within-level refinement. Initialization propagates coarse dual potentials across a recursive subsampling hierarchy to assign candidate edges. Refinement then iteratively inserts the largest dual violators in each row and column until the relative KKT residual meets a prescribed tolerance, while budgeted pruning ensures linear memory complexity. For exact-arithmetic refinement, we prove finite termination at a global optimum under a symbolic lexicographic rule. At the million-point scale, HELLO attains lower transport objectives with order-of-magnitude runtime improvements over strong baselines across feature dimensions from single digits to thousands. It further scales to 1.28 million samples per marginal in 8192 dimensions on a single H100, using 41.6 GiB peak GPU memory while satisfying a full relative KKT residual below $10^{-6}$. Beyond standard discrete OT, the framework supports general pairwise costs and serves as a scalable balanced-OT oracle for semi-discrete OT, Gromov--Wasserstein, unbalanced OT, and OT-based Flow Matching.

Dance-to-music (D2M) generation aims to synthesize music that is rhythmically and stylistically aligned with dance videos. A key challenge arises from the semantic mismatch between sparse dance cues, such as rhythm and style, and the dense information required for music composition, including structure, instrumentation, and expressive dynamics. Existing methods typically rely on these sparse cues and supervise only the final audio output, resulting in poorly learned music representations and generated music with limited musicality and structural coherence. To address these issues, we propose Curriculum-guided Multi-scale representation Alignment with scale-aware Optimal Transport (CMA-OT), a novel paradigm that leverages an external music expert to provide hierarchical supervision for the generator's latent features, bridging the semantic gap and enhancing representation learning. To effectively incorporate hierarchical supervision, we introduce a curriculum-guided multi-scale learning strategy that progressively transfers musical knowledge from the expert to the music generator, enabling stable and effective representation learning. Moreover, to accommodate the semantic and structural variations across different expert scales and achieve fine-grained alignment under temporal mismatch, we propose a scale-aware optimal transport alignment mechanism, which models soft correspondences between hierarchical expert representations and the generator's latent features. Extensive experiments on two datasets demonstrate that CMA-OT achieves state-of-the-art performance in rhythmic synchronization, perceptual quality, and overall music generation.

Optimal transport gives structured data a geometry, but exact evaluation is costly in large pairwise analyses that exploit relationships among distances. Learned surrogates are faster, but need not preserve this metric structure. We develop a Deep Sets surrogate for OT between variable-size weighted point clouds that enforces non-negativity, exchange symmetry, and zero self-distance, leaving the triangle inequality unconstrained. Applied to the Energy Mover's Distance between collider events in a particle physics application, the Metric-Aware Particle Flow Network achieves percent-level mean absolute percentage error while significantly improving inference throughput over other exact and approximate methods surveyed. The architectural constraints are found to improve properties that are not explicitly enforced: across $10^6$ held-out event triplets, triangle-inequality violations fall from 199 for a matched unconstrained network to 2, and the maximum from 149.5 to 5.8 GeV. These results demonstrate that targeted inductive biases can yield fast neural surrogates with substantially improved geometric fidelity.

Generalization under distribution shift remains a core challenge in modern machine learning, yet existing learning bound theory is limited to narrow, idealized settings and is non-estimable from samples. In this paper, we bridge the gap between theory and practical applications. We first show that existing definition of concept shift breaks when the source and target supports mismatch. Leveraging entropic optimal transport, we propose a key notion: $\gamma^{*}\!$-concept shifts, and derive a general error bound unifying covariate and $\gamma^{*}\!$-concept shifts, which applies to broad loss functions, label spaces, and stochastic labeling. We further develop estimators for these shifts with concentration guarantees, and the DataShifts algorithm, which can quantify distribution shifts and estimate the error bound in most applications - a rigorous and general tool for analyzing learning error under distribution shift.

Diffusion and flow-matching schedules control the signal and noise coefficients that mix data and noise along affine probability paths. Minimizing a kinetic action defined on coefficient paths, motivated by optimal transport, helps explain strong baselines but remains model-agnostic and ignores prediction error. Here we introduce a model-aware schedule construction based on fiberwise optimal transport. At a fixed time and state on the probability path, compatible signal/noise decompositions form an affine fiber. We define a fiberwise prediction risk by averaging optimal-transport costs between the true and predictor-induced decompositions within these fibers. On a fixed coefficient curve, combining this risk with coefficient-path kinetic action yields a closed-form optimal time allocation. This construction extends to general linear prediction targets, and the risk profile can be estimated from an early baseline checkpoint. We evaluate DDPMs and flow matching across prediction targets, training configurations, risk-estimation checkpoints, datasets, and architectures. Our model-aware schedules consistently outperform strong baselines, including a 38.6% relative FID reduction for flow matching on CIFAR-10 at 16 function evaluations. Each model-agnostic kinetic baseline determines its own kinetic reference coordinate. In these coordinates, fiberwise-risk profiles from independently trained models in different settings align closely after normalization to unit area. The resulting schedule deformations used in training also align, suggesting empirical universality across the evaluated models and settings. Pretrained-checkpoint diagnostics extend this normalized-risk agreement to larger conditional latent diffusion and 2-RF models. A frozen analytic allocation template retains most of the model-aware improvement without further risk estimation or model-specific fitting.

Estimating reliable cross-modality association is crucial to unsupervised visible-infrared person re-ID. While optimal transport is shown to be a practical solution for cross-modality association, it suffers from the rigidness of hard label assignment without considering the impact of cluster noise. Moreover, enforcing only cross-modality contrast is also suboptimal, as it fails to jointly optimize the similarity relation within and across modality. In this paper, we propose a novel framework for cross-modality learning by well exploitation of prototypes: First, instead of contrasting with cross-modality prototypes, we show that modality-unified prototypical contrast facilitates better modality invariance by jointly and simultaneously optimizing similarity relation within and across-modality. Taking self-prototype as a steady teacher, we further refine the instance-prototype online relation through prototype-guided self-distillation. The two components are optimized in a unified framework, leading to a simple yet effective model. On standard VI-ReID benchmarks, we perform extensive comparison and analysis, validating the effectiveness of our proposed method. Code is available at: https://github.com/Terminator8758/PoSeD.

Vision-language models such as CLIP embed images and text in a shared space, where modality-specific distributions often remain separated. Existing accounts connect this modality gap to initialization, contrastive dynamics, and information imbalance, while its distributional and pairwise contributions to retrieval remain unresolved. We introduce UOT-Gap, a training-free variational diagnostic that models frozen image and text embeddings with unbalanced entropic optimal transport (UOT). The UOT optimum separates transport, coupling complexity, and marginal mass variation; a complementary pair-aware residual compares observed image-caption pairs with the UOT soft matching. On Flickr8K and COCO-1K with frozen CLIP, OpenCLIP, and SigLIP encoders, caption degradation reduces Flickr8K Recall@1 from 0.559 to 0.003. Across six dataset-model conditions, the pair-aware residual tracks retrieval degradation with mean absolute Spearman 0.973, compared with 0.392 for the mean gap. The association remains stable across five random COCO-1K subsets at $0.954\pm0.026$, with a minimum of 0.943. UOT barycentric updates reduce the transport objective while degrading retrieval, distinguishing geometric objective descent from task improvement. These results establish UOT-Gap as a diagnostic for caption quality, modality alignment, and retrieval robustness.

Offline multi-objective optimization requires not only moving the objective vectors of candidate designs toward the Pareto front, but also distributing them effectively along it. Generative methods have recently emerged as a natural approach because they learn a distribution over feasible designs while allowing generation to be steered toward promising designs. Existing methods, however, largely retain classical sample-wise guidance strategies, leaving the distribution-level modeling capability of generative methods underused. We propose ParetoTransport, a training-free guidance method for pre-trained flow-matching models that explicitly specifies and refines a population-level distribution in objective space. ParetoTransport guides a flow-matching sampler to iteratively transport the empirical offline distribution toward the Pareto front, with Wasserstein matching to intermediate proxy distributions. This directly controls distributional displacement and mass allocation along the front. We establish a convergence result and demonstrate state-of-the-art performance on standard offline MOO benchmarks, extending recent evaluations beyond hypervolume to generational distance, inverted generational distance, and Wasserstein distance.

Current conditioned generative models heavily rely on score functions for guidance during training. When the generative model is a diffusion process with a singular diffusion coefficient and the underlying (conditional) densities either do not exist or are not smooth, we use causal optimal transport to define \emph{approximate} loss functions that identify a minimum-entropy control for guidance under minimal assumptions. Our approach relies on causal optimal transport and its characterization through the predictable representation property of (conditioned) diffusion processes whose associated martingale problem is well posed, à la Üstünel.

Infrared-visible object detection (IVOD) integrates complementary evidence from visible and infrared sensors for reliable perception in challenging scenes. In practice, sensors may fail or drop frames, leaving one modality unavailable or intermittent. Existing methods for IVOD assume both modalities are always present, and fixed fusion collapses when one stream is missing. Furthermore, it remains a critical challenge to reliably estimate semantic correlation across heterogeneous modalities, especially under spectral distribution discrepancy. We present FlexibleFusion, a unified and adaptive method that flexibly allocates integration pathways and fusion strength, operating seamlessly across complete and missing-modality regimes. At its core, the Modality-Aware Experts Collaboration (MAEC) mechanism selectively activates and aggregates cross-modal or intra-modal expert pathways. It allows cross-modal fusion when full modalities are available and falls back to self-fusion under missing conditions. Additionally, we design Residual Self-Paced Entropic Optimal Transport (RSPEOT) to align heterogeneous feature distributions from a transport perspective. Instead of relying on the fixed sparsity coefficient in standard entropic optimal transport (EOT), RSPEOT introduces a residual-driven self-paced update that prioritizes reliable matches and progressively refines harder ones. This design alleviates the additional optimization burden of standard EOT while preserving reliable semantic alignment. Comprehensive experiments under complete and missing-modality protocols show consistent performance across arbitrary modality configurations. Code will be released upon publication.

Prompt tuning provides a parameter-efficient way to adapt foundation models (FMs) by freezing the pretrained backbone and updating only a small set of learnable prompts. This property makes prompt tuning especially suitable for decentralized federated learning (DFL), where exchanging full-model updates can be prohibitively expensive. However, prompt tuning in DFL introduces new challenges. Prompt sets learned from heterogeneous local data may not be index-wise aligned, making standard decentralized averaging unsuitable. In addition, the algorithm should be theoretically guaranteed to achieve consensus and make progress toward the shared objective. In this work, we provide the first study of prompt tuning in DFL. We formulate decentralized prompt tuning as a Wasserstein-based optimization problem over prompt measures, which captures the set-valued structure of prompts. We then propose D-FROST, an optimal-transport-based (OT-based) decentralized prompt-tuning algorithm that merges neighborhood prompts into compact representative prompt sets through transportation-based matching. We further analyze D-FROST by bounding the Wasserstein consensus error across clients, and establishing convergence of the network-level prompt barycenter to a neighborhood of stationarity. Experiments under heterogeneous client data demonstrate the effectiveness of D-FROST for decentralized prompt tuning.

Flow Matching provides an efficient generative prior for image restoration by learning continuous transport between source and data distributions. However, existing methods typically incorporate measurement constraints through local corrections. Such corrections may disrupt the source-clean endpoint coupling implicitly encoded by the pretrained flow, making the corrected endpoint pair incompatible with the current state. To address this issue, we propose ReBridge-Flow, a posterior bridge re-coupling method. Specifically, given the current state, ReBridge-Flow first decodes the corresponding local source and clean endpoints. It then incorporates measurement information through clean-side anchoring and synchronously re-couples the source endpoint, yielding a measurement-aware endpoint pair with improved local bridge compatibility. The re-coupled endpoints further define a posterior-informed transport direction for advancing the sampling process. We also introduce the Posterior Bridge Defect, which jointly characterizes measurement error, deviation from the flow prior, and bridge mismatch, and leads to explicit updates for clean-side anchoring and source-side re-coupling. Extensive experiments on multiple natural and medical image restoration tasks demonstrate that ReBridge-Flow effectively alleviates bridge mismatch and improves the structural consistency of restored images.

A transformer carries each token's state from layer to layer, and the whole vocabulary carried together forms a cloud that moves with depth. We ask whether a trained network moves this cloud the way optimal transport would: at the cheapest cost, and along the map that pairs each token with its optimal destination. We measure both on Pythia-160m and Pythia-410m, with an exact assignment between consecutive layer clouds, a measured sampling floor, calibration on couplings known to be optimal, and a split of the cost into the common shift of the cloud and the token-specific moves. At the last layer, both models move their tokens where the optimal-transport map sends them, at the optimal cost for Pythia-410m and slightly above it for Pythia-160m. At the first layer they do not. In between, single layers can be judged on cost at only two of ten transitions, and blocks of several layers move the cloud at close to the optimal cost. The agreement at the last layer is much weaker at initialisation (0.64 against 0.86) and grows with training.

Unsupervised action segmentation is a challenging task. It involves finding action categories and boundaries in videos without labels. Existing Optimal Transport (OT) methods use global constraints. This causes them to overlook the use of local information. Furthermore, existing Optimal transport architectures are prone to confirmation bias because they overly trust the pseudo-labels they generate. This causes models to learn from noise in the early training stages. To address these issues, we propose FIS-OT. It is a novel Feature-Induced Structured Optimal Transport framework. First, we introduce a Feature Enhanced Generator (FEG) module. It serves as an internal regularizer. By using triplet loss, FEG captures local consistency. It provides robust supervision that is independent of noisy pseudo-labels. Second, we propose a Feature-Induced Residual Structural Prior. This combines a fixed temporal backbone with dynamic feature similarities. This design ensures temporal continuity. It also allows the solver to adapt to complex action structures. Finally, we establish a cyclic optimization loop. This aligns local feature learning with global structural alignment. Extensive experiments on the three datasets show the effectiveness of our method.

Geometry-conditioned multi-view diffusion enables high-quality 3D texture generation, but its repeated per-view denoiser evaluations introduce substantial computational cost. Existing training-free accelerators primarily exploit temporal redundancy by reusing computation across denoising steps. In multi-view texturing, however, skipping a step also removes the cross-view interaction that continually aligns different observations of the same surface, leading to rapidly degraded consistency and fidelity. Our analysis identifies a complementary source of redundancy: although intermediate features remain view-specific, geometrically corresponding surface points exhibit transferable evolution in their predicted clean signals. Based on this observation, we introduce \gc{}, a training-free plugin that evaluates a rotating subset of anchor views and transports their geometry-aligned per-step $\xz$ updates to the remaining views. Periodic full-view computation controls accumulated error, while sampler-consistent reconstruction preserves the denoising trajectory. \gc{} requires neither retraining nor architectural modification and uses the position maps already available in geometry-conditioned texturing pipelines. Across Hunyuan3D-2.1, SyncMVD, and MVPainter, \gc{} achieves a stronger speed--fidelity trade-off than temporal caches and step reduction at operating points above $2\times$. On Hunyuan3D-2.1, it delivers a $2.21\times$ denoiser-loop speedup with an MV-LPIPS of 0.0293 and an MV-PSNR of 33.60 dB, providing the best fidelity among all tested methods above $2\times$. The same transferred configuration reaches the highest speedup and lowest FLOPs on SyncMVD, while \gc{} achieves the lowest FLOPs and best fidelity among the accelerated methods on MVPainter. These results establish cross-view geometry as an effective acceleration axis for multi-view texture diffusion.

While 3D Vision-Language Models (3D VLMs) have demonstrated remarkable spatial reasoning capabilities, they suffer from massive visual token counts that create severe computational bottlenecks during inference. Existing token pruning methods primarily rely on diversity-based selection, discarding similar tokens to maximize dispersion. However, in 3D environments, this approach frequently drops representative prototype tokens in favor of outliers, breaking the multi-view consistencies and geometric structures essential for spatial reasoning. In this paper, we propose a paradigm shift for 3D VLM token pruning: from maximizing diversity to preserving visual evidence coverage. We introduce CoverPrune, a training-free framework that formulates inference-time token pruning as an Optimal Transport (OT) problem. To overcome the intractable combinatorial subset selection inherent in this formulation, we design the Feature-Spatial-Temporal (FST) transport cost and target capacity, along with an efficient Spatial-Guided Greedy Selection (SGS) algorithm to approximate the OT objective. Furthermore, we propose CoverPrune-Lite, an accelerated variant utilizing spatially structured local matching for minimal overhead. Extensive experiments across multiple 3D visual-spatial reasoning benchmarks demonstrate that our methods achieve state-of-the-art token efficiency, maintaining robust reasoning performance even under highly aggressive pruning budgets. Visit our project website at https://github.com/Brucess/CoverPrune.

We develop the statistical and algorithmic theory of inverse optimal transport (IOT) under the feature-parameterized cost C_theta(i,j) = -theta^T phi(i,j). The core technical contribution is the Sinkhorn linearization -- the implicit-function sensitivity of the entropic OT plan to the cost -- together with its spectral proxy, a formula that is spectrally exact yet geometrically transparent. The restricted Hessian on the tangent space satisfies the spectral sandwich (pi_min/epsilon) I <= H_T^{-1} <= (pi_max/epsilon) I, yielding the single core bound sigma_min >= (pi_min/(a_max epsilon)) sqrt(lambda_min(Sigma)) that drives the entire theory. On this core we establish four theorems and one observation. T1 (identifiability): theta is globally injective on the quotient of the gauge kernel, with dimension bound F <= (K-1)^2. T2 (sparsistency): the l1-penalized estimator recovers the true support under irrepresentability and score concentration, with exponential failure probability. T3 (well-posedness): the feature-moment map M(theta) = Phi^T x_theta is strongly monotone, and the inverse is Lipschitz with constant L <= epsilon ||Phi^T S_a||_op / (pi_min lambda_min(Sigma)). T4 (convergence): local strong convexity with mu >= pi_min^2 lambda_min(Sigma) / epsilon^2 guarantees monotone gradient descent convergence. O5 (misspecification): the estimator converges to the OT-model projection of the truth; the Holder continuity of the projection map is assessed numerically, yielding setting-dependent empirical exponents alpha_eff in (0,1).

Distribution steering seeks feedback laws that drive the state law of a dynamical system between prescribed initial and terminal distributions. Optimal transport provides a natural geometric approach, but its implementation generally requires a transport map or coupling in the full state space. Sliced optimal transport avoids this full-dimensional construction through one-dimensional projections. Yet, the resulting projected maps specify only directional displacements and do not by themselves prescribe a realizable feedback law. To this end, we develop a finite-horizon control framework based on sliced optimal transport. At each sampling instant, a projected optimal transport map defines a directional terminal condition, whose minimum-energy realization yields a randomized single-direction controller. Averaging over projection directions gives a deterministic sliced feedback. For the single-integrator dynamics, the averaged feedback makes the sliced Wasserstein distance to the target non-increasing. For Gaussian endpoint laws, it is affine, preserves Gaussianity, and steers the mean and covariance to their prescribed terminal values. We further identify a law-dependent gain that yields linear decay of the sliced Wasserstein distance together with an explicit characterization of the control energy. We also prove that the randomized controller converges to the averaged sliced flow as the sampling period vanishes. Finally, we extend the construction to linear dynamical systems. Reachability-normalized coordinates allow instantaneous realization of the sliced velocity for uniformly fully actuated systems, while local controllability Gramians provide exact finite-step realization for general controllable systems. Numerical examples illustrate the resulting distributional flows.

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.

Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces. However, we reveal that existing learnable projection mechanisms cannot ensure stable and balanced assignments from observation points to latent tokens, causing some latent tokens to be over-assigned while others remain underutilized. This limitation further restricts the design of hierarchical architectures, as assignment imbalance is continuously inherited and amplified across latent spaces, eventually causing severe token collapse in deeper spaces. To address these issues, we propose MoNo (Multiscale Optimal Transport Neural Operator), a progressive multiscale neural operator that efficiently solves PDEs on general geometries through stable latent-space construction. At its core is CoTAP (Cross-scale Optimal Transport Assignment and Projection), a novel latent-space construction method that formulates cross-space assignment between adjacent spaces as an entropy-regularized optimal transport problem, thereby constructing balanced bidirectional projections and stable latent spaces. CoTAP also ensures stable information transfer across multiple latent spaces, further enabling multiscale architectures on general geometries, which in turn support more efficient learning of long-range physical interactions. Extensive experiments demonstrate that MoNo outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency. Code is available at https://github.com/ZijiangY1116/MoNo.

Existing unified 4D reconstruction and point tracking approaches typically rely on heuristic interpolations or just predict at integer timestamps, lacking kinematic coherence and failing to model dynamics at any arbitrary timestamp. In this paper, we propose Uni4R, a framework that unifies these tasks by learning continuous velocity fields through the synergy of Optimal Transport (OT) and Ordinary Differential Equation (ODE). Importantly, this continuous velocity field acts as a kinematic prior that mutually benefits both 4D reconstruction and point tracking. Specifically, we propose the Flow Matching Guided Decoder (FMGD). A global velocity branch first extracts anchor features that capture the global dynamic state of the sequence. Then, FMGD leverages Flow Matching (FM) theory to formulate a probability path defined by OT on the anchor feature manifold, instantiating it as FM-guided velocity features for velocity prediction. This establishes a robust kinematic inductive bias. Meanwhile, a point reconstruction branch provides geometric features. The local velocity prediction module then joint above features and time embeddings, to decode velocities at arbitrary timestamps. To overcome the absence of high-quality ground-truth velocities in fractional frames, we propose an integral-consistency training strategy. This strategy uses an ODE solver to integrate velocities to recover target pointmaps, enabling the model to be supervised end-to-end directly from integer timestamps. Experimental results demonstrate that Uni4R achieves SOTA performance in both 4D reconstruction and point tracking, and achieves SOTA in our new kinematics-aware benchmark at continuous time.

Individual fish re-identification (ReID) is a fine-grained recognition problem in which identity-discriminative cues are often localized to specific body regions rather than distributed uniformly across the animal. Nevertheless, recent CLIP-based ReID methods rely predominantly on global image-text alignment, allowing background and weakly discriminative regions to contribute to cross-modal supervision. We propose a selective local vision-language alignment framework that establishes localized correspondences between visual patch embeddings and multiple identity-aware prompt embeddings through Partial Optimal Transport (POT). Rather than enforcing exhaustive correspondence, POT enables selective matching between visual patches and prompt embeddings, allowing the model to emphasize the strongest cross-modal correspondences while avoiding forced alignment of weakly matching regions, thereby yielding more discriminative visual representations for retrieval. The framework is trained end-to-end, while only the adapted visual encoder is retained during inference. Experiments on the longitudinal Symphodus melops dataset demonstrate consistent improvements over recent CLIP-based ReID methods under both closed-set and open-set evaluation protocols. Additional evaluations on other datasets further demonstrate the generalization capability of the proposed method across diverse marine ReID benchmarks.

Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is $N$-cyclically monotone, where $N$ is the batch size. Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.

The development of deep learning over the past decade has revolutionized medical imaging segmentation, allowing the extraction of precise descriptors from large volumes to characterize pathologies. Data augmentation is a technique widely regarded as a way to improve model training. It includes simple transformations like spatial operations or intensity modifications, but also more advanced synthesis techniques. Their goal is to generate new realistic samples from an existing dataset to diversify the images used during training. Among them, several propose different mixing strategies to combine real samples. However, one of their major shortcomings is to yield limited variability in terms of generated lesion shapes and locations. In this work, we introduce a novel image synthesis method, called OTLesMix, that leverages Wasserstein barycenter and optimal transport plan to generate realistic and diverse samples. We evaluated our method on three brain lesion segmentation tasks, on which it improves the Dice score compared to a model trained without synthetic data by 2.9 to 6.6 points, and outperforms state-of-the-art mix-based methods.

Optimal transport (OT) has emerged as an effective framework for unsupervised action segmentation. Yet, in existing OT-based methods, the latent action prototypes that define the OT costs are not re-estimated from the refined frame geometry. Instead, they evolve solely through gradients from the pseudo-label loss. We identify this \emph{representation--prototype inconsistency} as a central bottleneck, particularly around ambiguous transitions and for short or infrequent actions. To address this issue, we build on the recently introduced CLOT, which refines frame embeddings based on estimated segment embeddings, and further re-estimates the action prototypes from the refined frame embeddings. Specifically, we introduce a graph-constrained module that regularizes the OT-refined frame and segment representations by preserving the local neighborhood geometry of the encoder output. An action-embedding refinement step then periodically re-anchors the prototypes to this stabilized representation geometry. We study two instantiations that share the same backbone, graph module, and objective: D-CLOT updates the prototypes using $k$-means, whereas D-CLOT$_{B}$ updates them as OT barycenters weighted by the refined transport plan, yielding an assignment-aware prototype update consistent with the current transport geometry. Across five established benchmarks, both variants improve segment-level quality over CLOT, with per-video gains of up to $+12.7$ F1 and $+10.2$ mIoU (YTI) and activity-level gains of up to $+8.9$ F1 (FS-Eval). We further establish the first unsupervised action-segmentation baseline on Assembly101, a procedural and substantially more fine-grained benchmark than those commonly used in prior work. Extensive ablations and sensitivity analyses demonstrate that the two refinement mechanisms are complementary and robust.

We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou--Brenier form for the chosen exponent $p$. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou--Brenier optimality system and recovers the corresponding $p$-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.

We study the problem of aligning data from multiple modalities into a shared representation space, focusing on settings where strong pretrained unimodal encoders are available but cross-modal paired data are scarce. We propose a structure-preserving alignment framework, joint kernel entropic Gromov--Wasserstein Optimal Transport (JK-EGW), which maps multiple modalities into a common latent space by minimizing a quadratic optimal transport objective. JK-EGW leverages fine-grained similarity relationships within and across modalities to construct a global affinity kernel instead of relying on raw feature-space distances. Our framework naturally provides explicit control over the geometry and distribution of the latent embedding. On the theory side, we establish parametric sample complexity rate of $n^{-1/2}$, matching the corresponding rates for standard, entropic and Gromov--Wasserstein optimal transport. On the algorithmic side, we derive a scalable alternating procedure to solve JK-EGW with entropic optimal transport (EOT) updates through a low-rank kernel approximation and a variational lifting. This lifting scheme effectively relieves the burden of a quadratic objective, and allowing us to take the advantage of existing EOT solvers. Empirically, we focus on post-hoc alignment of embeddings from pretrained encoders in data-scarce regimes, and show that our proposed method achieves improved multimodal retrieval performance compared to existing alignment baselines.

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 $ε$-Adaptive Selection ($ε$-AS). Extensive experiments on six public datasets and multiple annotation budgets show that $ε$-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

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 \textit{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. In contrast, under suitable compactness and uniform-integrability assumptions, iterative \textit{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 \textit{c}-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension $d \ge 3$ and a nearly parametric rate for $d=1,2$.

High-polyphony symbolic music is increasingly used in generation, analysis, and arrangement, yet many downstream tasks require bounded representations with fixed tracks or slots. Converting richly orchestrated scores into compact forms is therefore necessary, but existing approaches relying on heuristic simplification or generic representation-space reduction often fail to preserve structural roles, orchestration compatibility, and playability under strict budgets. To address the issue, this study reformulates the compression problem as a fixed-budget structured routing problem and proposes Unbalanced Optimal Transport for Information Routing (UOT-IR), a training-free framework based on constrained unbalanced optimal transport. UOT-IR combines an orchestration prior, adaptive marginal relaxation, temporal decoding, and playability-aware projection to produce compact and musically coherent bounded representations. This work further studies two practical settings under the same slot budget: template standardization, which maps each input to a predefined bounded template, and adaptive preservation, which retains representative content without assuming an external template. Experiments on the SymphonyNet corpus show that UOT-IR delivers strong overall performance across both settings, including the best Note-F1 in adaptive preservation (0.9120), together with the lowest structural cost (14.7165) and bad structural confusion rate (0.3406) in template standardization. This work establishes a principled paradigm for fixed-budget symbolic music compression, offering a practical path toward compact, structured, and musically coherent symbolic representations.

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 $η$ can be completed by sparse target-cell residuals into an exact-marginal value-certified sampler with scalar certificate $W_1(μ,ν)\leq U\leq W_1(μ,ν)+2Δ$, where $Δ$ is the public target-partition diameter. The certificate accuracy is controlled by $Δ$ alone. The field error $η$ controls residual communication under a cell-margin condition; without a margin, $η$ alone does not bound residuals. We instantiate the compiler via adaptive local-affine and tensor-product spline codes with $d(m+1)^db$ 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 $Ω(\varepsilon^{-2d/(d+4)})$ 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.

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.

Control valve stiction is a common cause of unwanted oscillations and poor control-loop performance in industrial processes. Data-driven methods can automatically detect stiction, but models trained purely on simulated data often struggle to generalize to real industrial control loops due to domain shift. To bridge this gap, this work propose a novel stiction detection methodology that combines optimal transport (OT) imaging technique with deep correlation alignment (Deep CORAL) algorithm. Closed loop signals: controller output and process variable are converted into two-dimensional OT images. These images capture the dynamic behaviour of control loops. The proposed methodology includes a convolutional neural network (CNN) encoder (or feature extractor) trained to learn domain-invariant representations by optimizing a combined objective: a cross-entropy loss on labeled simulation data and a Deep CORAL (covariance-alignment) loss between simulation data and unlabeled target-domain industrial data. Downstream classifiers trained on the domain-invariant target features were evaluated on an independent test set of 20 benchmark loops from industrial stiction data benchmark. The proposed methodology successfully diagnosed 18 out of the 20 loops and achieved100% recall across all 13 stiction cases, an accuracy of 90.00% and an F1-score of 92.86%. Compared to standard baseline approach (hand-crafted features-based method), the proposed methodology significantly mitigates domain shift, providing robust, highly reliable stiction detection for real-world industrial control loops.

We present a privacy-preserving framework for synthetic lung CT slice generation developed for the Image-CLEFmed GANs 2026 challenge. The approach combines Optimal Transport Conditional Flow Matching with privacy-oriented training and a post-generation "Supervisor" pipeline that filters generated candidates in learned geometric latent spaces using autoencoder embeddings, Determinantal Point Processes, and Stein Kernel Thinning. Official results show a strong realism-privacy trade-off, with the best-performing model achieving a Privacy Preservation Score of 0.549 and competitive visual fidelity with an FID of 0.3290. While the proposed geometric filtering substantially reduces nearest-neighbor memorization and membership-inference leakage, persistent patient re-identification scores indicate that preventing direct image copying is not sufficient to remove deeper patient-specific anatomical identity, highlighting an important frontier for future privacy-preserving medical image generation.

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.

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.

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.

Linear Independent Component Analysis (ICA) recovers jointly independent source signals from their linear mixtures. To achieve this, classical ICA algorithms attempt to maximize non-Gaussianity, measured by negentropy, which is linked to independence by information theory. Because exact negentropy optimization is intractable, they rely on proxy contrast functions, such as fourth-order cumulants, and parametric log-likelihoods. We propose instead to measure non-Gaussianity using the squared Wasserstein distance $W_2^2$ to a standard Gaussian. We prove that the Wasserstein distance between a standard normal distribution and linear projections of the data is maximized when the projection recovers an independent component. Based on this observation, we propose the OT-ICA algorithm which finds this projection by gradient-based optimization. Empirical evaluation on simulated data shows that OT-ICA outperforms proxy-based methods for different distributions of the latent variables. Application to EEG artifact removal and econometric price discovery confirm OT-ICA can be used for applied ICA tasks without distributional assumptions.

The analysis of Multivariate Time Series (MTS) plays an important role in a lot of real-world practical applications, but it still remains some challenging problem about capturing multi-granularity structural patterns and suppressing noise appropriately. Multi-Scale Convolution with Optimal Transport Attention (MSC-OT) is proposed in this paper. MSC-OT is a useful architecture to optimize the attention mechanism. It combines multi-scale convolution with Sinkhorn optimal transport method based on inverted embedding. The inverted embedding approach embeds each variable as a token and allows the model to capture cross-variate relationships better. MSC-OT consists of two part: (1) Multi-Scale Convolution Enhancement, that applies multi-scale convolutions to attention score matrices based on inverted embedding, capturing local structural patterns in the variate-interaction space induced by compressed temporal representations; (2) Sinkhorn Optimal Transport Regularization, that formulates attention computation as an optimal transport problem and employs iterative matrix scaling to ensure balanced information flow across variates. Adaptive Fusion Strategy utilizes softmax-normalized learnable weights to dynamically combine base attention, convolution-enhanced, and OT-regularized scores. Experiments on widely-used datasets, including ETT, Electricity, Traffic, Solar-Energy, and Exchange-Rate, show that MSC-OT achieves well performance in both short-term and long-term forecasting tasks. Ablation experiments further validate the effectiveness of each proposed component and their synergistic contributions to improving prediction accuracy for multivariate time series forecasting.

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: \href{https://github.com/YZH0905/CAOT-STC}{https://github.com/YZH0905/CAOT-STC}.

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.

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.

Large language model (LLM)-based audio-visual speech recognition (LLM-AVSR) has recently demonstrated strong robustness in adverse acoustic environments by leveraging complementary audio and visual information. Existing approaches typically employ independently pretrained acoustic and visual encoders, whose outputs are projected and fused as soft prompts to condition an LLM for speech recognition. However, most methods perform multimodal fusion without explicitly addressing the representational discrepancy between audio, visual and text modalities, potentially limiting the effectiveness of cross-modal integration. In this paper, we propose an optimal transport (OT)-based semantic alignment framework for LLM-AVSR. The proposed method explicitly bridges the modality gap by aligning the acoustic and visual representations with reference to the linguistic embedding space of the LLM before multimodal fusion. Specifically, OT is used to estimate probabilistic coupling matrices that characterize structured correspondences between modality-specific features and linguistic embeddings. The resulting OT couplings are further utilized as soft pseudo-labels to supervise contrastive learning, encouraging the extraction of semantically coherent and cross-modal consistent audio-visual representations. By anchoring multimodal features to the linguistic space of the LLM, the proposed framework facilitates more effective multimodal fusion and decoding. We implement the proposed framework using a Whisper-based acoustic encoder, an AV-HuBERT-based visual encoder, and a LLaMA3.2-3B decoder. Experiments conducted on the LRS3-TED benchmark demonstrate consistent improvements over strong baselines and achieve state-of-the-art performance under both clean and noisy evaluation conditions across a wide range of signal-to-noise ratios (SNRs).

Color transfer aims to align the color distribution of a source image with that of a reference image while preserving structural and semantic consistency. However, existing methods often suffer from inaccurate global mapping, semantic misalignment, and visual artifacts. To address these issues, we propose ColorFM, an optimization-to-learning framework. ColorFM connects online optimization to offline inference by reformulating color transfer as the transport of pixel distributions along velocity fields via Flow Matching. Specifically, we introduce ColorFM-O, an instance-specific optimization scheme that fits the velocity field through hierarchical color coupling guided by semantic priors. By numerically integrating the induced flow trajectories, ColorFM-O produces precise and semantically consistent color transfer results, while generating high-quality paired data as pseudo-supervision. Building upon this, we design ColorFM-L, an efficient feed-forward model trained on the generated pairs. Through implicit state modeling, ColorFM-L extracts deep semantic features to predict flow parameters for bidirectional linearized transport, ensuring accurate color transfer. Extensive experiments demonstrate that ColorFM-L outperforms state-of-the-art methods in visual quality, structural fidelity, and semantic consistency, successfully combining the accuracy of optimization with the speed of feed-forward inference.

This paper introduces Diffusion Semi-Relaxed Fused Gromov-Wasserstein (DsrFGW), a novel method for graph comparison that unifies node features and structural connectivity through optimal transport. While traditional Gromov-Wasserstein and semi-relaxed variants (srGW, srFGW) capture graph structure, they often struggle with sparse, noisy, or partially observed graphs. Inspired by Graph Diffusion Distance, which posits graphs are similar if they enable similar information transmission patterns, DsrFGW incorporates diffusion processes allowing information propagation across nodes, capturing local and global structural patterns while reducing sensitivity to noise or missing edges. An extensive evaluation on 36 synthetic pairwise graph matching tasks (easy, medium, hard) demonstrates consistent superiority over srFGW, achieving accuracy improvements of 0-20 percentage points and dramatic Adjusted Rand Index (ARI) gains: in medium-difficulty scenarios, srFGW often achieves negative ARI (worse than random) while DsrFGW offers better performance in terms of both internal and external clustering quality measures (i.e., Adjusted Rank Index and Accuracy with respect to the true underlying clusters, respectively). Even under severe noise, DsrFGW improves clustering quality in 92% of the synthetic tasks with optimal diffusion scales adapting to problem difficulty, establishing DsrFGW as a robust framework for graph comparison under structural uncertainty.

Diffusion and flow policies have recently demonstrated remarkable performance in robotic applications by accurately capturing multimodal robot trajectory distributions, especially in the context of vision language action (VLA) models. However, high quality policy performance also requires fast inference and high quality demonstrations, which are often hard to get. Lack of these leads to suboptimal policy behaviors and failure under distribution shifts. In this work we address the problem of fine-tuning and accelerating suboptimal flow-based policies using the robot's experience through RL post-training. We introduce Optimal Transport Q-Learning (OTQL), a new method for finetuning flow policies using advantage weighted conditional optimal transport flow matching. OTQL can finetune and accelerate flows with an interaction budget of 50-60 episodes while avoiding computationally expensive distillation in simulation and real-world robot tasks. Our results show that OTQL post-trains flow policies using the robot's own experience, increasing average success percentage of single-task policies from 36% to 86% and of a pre-trained VLA from 38% to 76% while reducing the number of inference steps per action generation by 70%.

This paper concerns the problem of unsupervised temporal action segmentation for long, untrimmed videos. Recent successful approaches follow a joint representation learning and clustering paradigm, where optimal transport (OT) is adopted to produce pseudo labels for learning frame representations. These approaches alternate between estimating pseudo labels using OT and optimizing the parameters with gradient descent during training, where OT is used for obtaining the final temporal action segmentation. A major limitation of these works is that they learn a deterministic embedding for frame representations. The iterative procedure between learning deterministic embeddings based on pseudo labels and estimating pseudo labels from the learned embedding can thus get quickly stuck in a local optimum. As an alternative, we thus propose to learn a probabilistic embedding for frame representations. The embeddings are modeled by Gaussian distributions and we sample from the distributions before estimating the pseudo labels. We evaluate our approach on several challenging temporal action segmentation datasets and achieve results comparable to, and in some cases, better than the state of the art. Compared to baselines with deterministic embeddings, our approach improves MoF up to 20.7% and F1-score up to 19.0%. Our code is available at https://github.com/derkbreeze/PEOT.

Recurrent representations are trajectories, but representation geometry is often measured from static snapshots. We develop finite-lag operator geometry for recurrent hidden states from observed source-successor pairs $(X_t,X_{t+Δ})$. The primitive is the conditional transport law $Q_Δ(dy\mid x)$, estimated by a dense Gaussian source-smoothing operator. From this directed finite-lag law we derive a source-centered transport tensor $G_Δ$, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation $W_Δ^ρ$, which summarizes directed lagged flow. We prove affine covariance with explicit metric dependence of scalar summaries, dense estimator stability on bounded trajectory clouds, and a finite-lag separation result showing that source-centered transport detects deterministic recurrent motion not recorded by infinitesimal carre-du-champ geometry. A linear-Gaussian closed form calibrates the quantities in terms of the update $A_Δ$, source covariance, and innovation covariance. Controlled experiments validate the decomposition, circulation, covariance, and stability predictions. In performance matched repeat-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace, while coherent displacement fraction is metric and resolution dependent.

Self-supervised Continual Graph Learning (CGL) aims to successively learn from a graph sequence with different tasks without label supervision - a paradigm that has attracted widespread attention. Most existing self-supervised CGL methods rely on instance-level consistency objectives that enforce stability of individual node (or node-pair) embeddings. Due to optimizing nodes in isolation, these methods fail to maintain global relational structure, causing inter-node correspondences to progressively distort under continual learning. To this end, we propose a novel Structure-Aware Optimal Transport (SAOT) framework that explicitly captures and preserves relational structure within graph representations across sequential tasks. Specifically, SAOT leverages optimal transport theory to capture global inter-node correspondences, thereby facilitating and enhancing graph representation learning. Simultaneously, SAOT incorporates a cross-task knowledge distillation mechanism to preserve the previous structural knowledge. Extensive experiments on four CGL benchmark datasets demonstrate that SAOT outperforms existing self-supervised baselines. In particular, SAOT achieves significant performance gains, improving average accuracy by up to 5% on CoraFull-CL and over 15% on Products-CL compared with state-of-the-art methods in the Class-IL setting.

We propose OTCache, a training-free framework for accelerating diffusion sampling via caching schedule prediction. Existing graph-based caching methods reduce redundant computation by optimizing shortest-path objectives, but rely on an additive independence assumption, which often breaks down in the low NFE regime. To address this issue, OTCache models caching schedules across inference budgets as a smooth evolution in policy space, inspired by Optimal Transport (OT). The framework consists of three stages: (1) obtaining a high-fidelity \textbf{reference schedule} using a graph-based caching method under a conservative budget; (2) performing a lightweight anchor search under an extreme low-budget setting via Optuna optimization with an end-to-end perceptual objective; and (3) predicting schedules for target budgets via quantile interpolation between the reference and anchor policies using continuous warping representations. Experiments on FLUX.1 [dev], Qwen-Image, and HunyuanVideo show that OTCache achieves 4.5x, 4.7x, and 3.66x acceleration, respectively, while consistently improving generation fidelity over state-of-the-art caching baselines. This work provides a new perspective on accelerating diffusion models through Optimal-Transport-inspired schedule modeling. Code:https://github.com/UnicomAI/OTCache