Daily Papers

Destructive single-cell measurements provide unpaired population snapshots rather than observations of the same cells across conditions. Local cell states and transition requests may also be insufficient to distinguish responses across source populations. We introduce TempoBridge, a common source-conditioned transport formulation for temporal, genetic, and chemical population transitions. Source cells initialize latent transport and provide a fixed empirical population summary. The velocity field receives this summary alongside the evolving cell state, flow time, and a structured transition descriptor. Minibatch optimal transport (OT) supplies couplings only for conditional flow-matching training paths; inference requires neither target expression nor OT computation. On held-out donors, TempoBridge achieves an Energy distance of 0.129 versus 0.144 for scGen. Genetic mean-expression L_2 error is 2.261 versus 3.156 for scGPT-scratch under Seen 2/2. On held-out compounds, condition-averaged drug-effect correlation is 0.598 versus 0.561 for the CellFlow adapter. Temporal ablations show higher mean distributional error after removing source context, replacing optimal transport with random pairing, or replacing flow matching with static residual regression. Together, these results demonstrate the predictive utility of a common source-conditioned transport formulation across held-out donors, gene combinations, and compounds.

Learning a transport model that maps a source distribution to a target distribution is a canonical problem in machine learning, but scientific applications increasingly require models that can generalize to source and target distributions unseen during training. We introduce distribution-conditioned transport (DCT), a framework that conditions transport maps on learned embeddings of source and target distributions, enabling generalization to unseen distribution pairs. DCT also allows semi-supervised learning for distributional forecasting problems: because it learns from arbitrary distribution pairs, it can leverage distributions observed at only one condition to improve transport prediction. DCT is agnostic to the underlying transport mechanism, supporting models ranging from flow matching to distributional divergence-based models (e.g. Wasserstein, MMD). We demonstrate the practical performance benefits of DCT on synthetic benchmarks and four applications in biology: batch effect transfer in single-cell genomics, perturbation prediction from mass cytometry data, learning clonal transcriptional dynamics in hematopoiesis, and modeling T-cell receptor sequence evolution.

Recovering 3D human poses from a monocular camera view is a highly ill-posed problem due to the depth ambiguity. Earlier studies on 3D human pose lifting from 2D often contain incorrect-yet-overconfident 3D estimations. To mitigate the problem, emerging probabilistic approaches treat the 3D estimations as a distribution, taking into account the uncertainty measurement of the poses. Falling in a similar category, we proposed FMPose, a probabilistic 3D human pose estimation method based on the flow matching generative approach. Conditioned on the 2D cues, the flow matching scheme learns the optimal transport from a simple source distribution to the plausible 3D human pose distribution via continuous normalizing flows. The 2D lifting condition is modeled via graph convolutional networks, leveraging the learnable connections between human body joints as the graph structure for feature aggregation. While trade-offs between processing time and precision exist, already in the equal-accuracy comparison, FMPose exhibits significantly faster processing time than the diffusion model, and also offers another faster and more accurate configuration. Experimental results show major improvements of our FMPose over current state-of-the-art methods on two common benchmarks for 3D human pose estimation, namely Human3.6M, MPI-INF-3DHP. Additionally, FMPose shows competitive performance on the more challenging 3DPW dataset. The code implementation is available at https://github.com/cuongle1206/FMPose

We present a neural framework for learning conditional optimal transport (OT) maps between probability distributions. Our approach introduces a conditioning mechanism capable of processing both categorical and continuous conditioning variables simultaneously. At the core of our method lies a hypernetwork that generates transport layer parameters based on these inputs, creating adaptive mappings that outperform simpler conditioning methods. Comprehensive ablation studies demonstrate the superior performance of our method over baseline configurations. Furthermore, we showcase an application to global sensitivity analysis, offering high performance in computing OT-based sensitivity indices. This work advances the state-of-the-art in conditional optimal transport, enabling broader application of optimal transport principles to complex, high-dimensional domains such as generative modeling and black-box model explainability.

To measure the difference between two probability distributions, referred to as the source and target, respectively, we exploit both the chain rule and Bayes' theorem to construct conditional transport (CT), which is constituted by both a forward component and a backward one. The forward CT is the expected cost of moving a source data point to a target one, with their joint distribution defined by the product of the source probability density function (PDF) and a source-dependent conditional distribution, which is related to the target PDF via Bayes' theorem. The backward CT is defined by reversing the direction. The CT cost can be approximated by replacing the source and target PDFs with their discrete empirical distributions supported on mini-batches, making it amenable to implicit distributions and stochastic gradient descent-based optimization. When applied to train a generative model, CT is shown to strike a good balance between mode-covering and mode-seeking behaviors and strongly resist mode collapse. On a wide variety of benchmark datasets for generative modeling, substituting the default statistical distance of an existing generative adversarial network with CT is shown to consistently improve the performance. PyTorch code is provided.

We present Pix2Gif, a motion-guided diffusion model for image-to-GIF (video) generation. We tackle this problem differently by formulating the task as an image translation problem steered by text and motion magnitude prompts, as shown in teaser fig. To ensure that the model adheres to motion guidance, we propose a new motion-guided warping module to spatially transform the features of the source image conditioned on the two types of prompts. Furthermore, we introduce a perceptual loss to ensure the transformed feature map remains within the same space as the target image, ensuring content consistency and coherence. In preparation for the model training, we meticulously curated data by extracting coherent image frames from the TGIF video-caption dataset, which provides rich information about the temporal changes of subjects. After pretraining, we apply our model in a zero-shot manner to a number of video datasets. Extensive qualitative and quantitative experiments demonstrate the effectiveness of our model -- it not only captures the semantic prompt from text but also the spatial ones from motion guidance. We train all our models using a single node of 16xV100 GPUs. Code, dataset and models are made public at: https://hiteshk03.github.io/Pix2Gif/.

Conditional generative modeling aims to learn a conditional data distribution from samples containing data-condition pairs. For this, diffusion and flow-based methods have attained compelling results. These methods use a learned (flow) model to transport an initial standard Gaussian noise that ignores the condition to the conditional data distribution. The model is hence required to learn both mass transport and conditional injection. To ease the demand on the model, we propose Condition-Aware Reparameterization for Flow Matching (CAR-Flow) -- a lightweight, learned shift that conditions the source, the target, or both distributions. By relocating these distributions, CAR-Flow shortens the probability path the model must learn, leading to faster training in practice. On low-dimensional synthetic data, we visualize and quantify the effects of CAR. On higher-dimensional natural image data (ImageNet-256), equipping SiT-XL/2 with CAR-Flow reduces FID from 2.07 to 1.68, while introducing less than 0.6% additional parameters.

Optimal Transport (OT) has fueled machine learning (ML) across many domains. When paired data measurements (mu, nu) are coupled to covariates, a challenging conditional distribution learning setting arises. Existing approaches for learning a global transport map parameterized through a potentially unseen context utilize Neural OT and largely rely on Brenier's theorem. Here, we propose a first-of-its-kind quantum computing formulation for amortized optimization of contextualized transportation plans. We exploit a direct link between doubly stochastic matrices and unitary operators thus unravelling a natural connection between OT and quantum computation. We verify our method (QontOT) on synthetic and real data by predicting variations in cell type distributions conditioned on drug dosage. Importantly we conduct a 24-qubit hardware experiment on a task challenging for classical computers and report a performance that cannot be matched with our classical neural OT approach. In sum, this is a first step toward learning to predict contextualized transportation plans through quantum computing.

Minibatch optimal transport coupling straightens paths in unconditional flow matching. This leads to computationally less demanding inference as fewer integration steps and less complex numerical solvers can be employed when numerically solving an ordinary differential equation at test time. However, in the conditional setting, minibatch optimal transport falls short. This is because the default optimal transport mapping disregards conditions, resulting in a conditionally skewed prior distribution during training. In contrast, at test time, we have no access to the skewed prior, and instead sample from the full, unbiased prior distribution. This gap between training and testing leads to a subpar performance. To bridge this gap, we propose conditional optimal transport C^2OT that adds a conditional weighting term in the cost matrix when computing the optimal transport assignment. Experiments demonstrate that this simple fix works with both discrete and continuous conditions in 8gaussians-to-moons, CIFAR-10, ImageNet-32x32, and ImageNet-256x256. Our method performs better overall compared to the existing baselines across different function evaluation budgets. Code is available at https://hkchengrex.github.io/C2OT

Despite their impressive capabilities, current Text-to-Image (T2I) models remain prone to generating unsafe and toxic content. While activation steering offers a promising inference-time intervention, we observe that linear activation steering frequently degrades image quality when applied to benign prompts. To address this trade-off, we first construct SafeSteerDataset, a contrastive dataset containing 2300 safe and unsafe prompt pairs with high cosine similarity. Leveraging this data, we propose Conditioned Activation Transport (CAT), a framework that employs a geometry-based conditioning mechanism and nonlinear transport maps. By conditioning transport maps to activate only within unsafe activation regions, we minimize interference with benign queries. We validate our approach on two state-of-the-art architectures: Z-Image and Infinity. Experiments demonstrate that CAT generalizes effectively across these backbones, significantly reducing Attack Success Rate while maintaining image fidelity compared to unsteered generations. Warning: This paper contains potentially offensive text and images.

We investigate whether the Feed-Forward Network (FFN) sublayer in a decoder-only transformer can be replaced by an explicit learned memory graph while preserving the surrounding autoregressive architecture. The proposed Graph Memory Transformer (GMT) keeps causal self-attention intact, but replaces the usual per-token FFN transformation with a memory cell that routes token representations over a learned bank of centroids connected by a learned directed transition matrix. In the base GMT v7 instantiation studied here, each of 16 transformer blocks contains 128 centroids, a 128 * 128 edge matrix, gravitational source routing, token-conditioned target selection, and a gated displacement readout. The cell therefore returns movement from an estimated source memory state toward a target memory state, rather than a retrieved value. The resulting model is a fully decoder-only language model with 82.2M trainable parameters and no dense FFN sublayers, compared with a 103.0M-parameter dense GPT-style baseline used in the evaluation. The base v7 model trains stably and exposes centroid usage, transition structure, and source-to-target movement as directly inspectable quantities of the forward computation. It remains behind the larger dense baseline in validation loss and perplexity (3.5995/36.58 vs. 3.2903/26.85), while showing close zero-shot benchmark behavior under the evaluated setting. These results are not intended as a state-of-the-art claim; they support the viability and structural interpretability of replacing dense within-token transformation with graph-mediated memory navigation. Broader scaling, optimized kernels, and more extensive benchmark evaluation are left for subsequent work.

3D pose transfer aims to transfer the pose-style of a source mesh to a target character while preserving both the target's geometry and the source's pose characteristic. Existing methods are largely restricted to characters with similar structures and fail to generalize to category-free settings (e.g., transferring a humanoid's pose to a quadruped). The key challenge lies in the structural and transformation diversity inherent in distinct character types, which often leads to mismatched regions and poor transfer quality. To address these issues, we first construct a million-scale pose dataset across hundreds of distinct characters. We further propose MimiCAT, a cascade-transformer model designed for category-free 3D pose transfer. Instead of relying on strict one-to-one correspondence mappings, MimiCAT leverages semantic keypoint labels to learn a novel soft correspondence that enables flexible many-to-many matching across characters. The pose transfer is then formulated as a conditional generation process, in which the source transformations are first projected onto the target through soft correspondence matching and subsequently refined using shape-conditioned representations. Extensive qualitative and quantitative experiments demonstrate that MimiCAT generalizes plausible poses across diverse character morphologies, surpassing prior approaches restricted to narrow-category transfer (e.g., humanoid-to-humanoid).

Visual object tracking has gained promising progress in past decades. Most of the existing approaches focus on learning target representation in well-conditioned daytime data, while for the unconstrained real-world scenarios with adverse weather conditions, e.g. nighttime or foggy environment, the tremendous domain shift leads to significant performance degradation. In this paper, we propose UMDATrack, which is capable of maintaining high-quality target state prediction under various adverse weather conditions within a unified domain adaptation framework. Specifically, we first use a controllable scenario generator to synthesize a small amount of unlabeled videos (less than 2% frames in source daytime datasets) in multiple weather conditions under the guidance of different text prompts. Afterwards, we design a simple yet effective domain-customized adapter (DCA), allowing the target objects' representation to rapidly adapt to various weather conditions without redundant model updating. Furthermore, to enhance the localization consistency between source and target domains, we propose a target-aware confidence alignment module (TCA) following optimal transport theorem. Extensive experiments demonstrate that UMDATrack can surpass existing advanced visual trackers and lead new state-of-the-art performance by a significant margin. Our code is available at https://github.com/Z-Z188/UMDATrack.

Recent advances in text-to-image synthesis have led to large pretrained transformers with excellent capabilities to generate visualizations from a given text. However, these models are ill-suited for specialized tasks like story visualization, which requires an agent to produce a sequence of images given a corresponding sequence of captions, forming a narrative. Moreover, we find that the story visualization task fails to accommodate generalization to unseen plots and characters in new narratives. Hence, we first propose the task of story continuation, where the generated visual story is conditioned on a source image, allowing for better generalization to narratives with new characters. Then, we enhance or 'retro-fit' the pretrained text-to-image synthesis models with task-specific modules for (a) sequential image generation and (b) copying relevant elements from an initial frame. Then, we explore full-model finetuning, as well as prompt-based tuning for parameter-efficient adaptation, of the pre-trained model. We evaluate our approach StoryDALL-E on two existing datasets, PororoSV and FlintstonesSV, and introduce a new dataset DiDeMoSV collected from a video-captioning dataset. We also develop a model StoryGANc based on Generative Adversarial Networks (GAN) for story continuation, and compare it with the StoryDALL-E model to demonstrate the advantages of our approach. We show that our retro-fitting approach outperforms GAN-based models for story continuation and facilitates copying of visual elements from the source image, thereby improving continuity in the generated visual story. Finally, our analysis suggests that pretrained transformers struggle to comprehend narratives containing several characters. Overall, our work demonstrates that pretrained text-to-image synthesis models can be adapted for complex and low-resource tasks like story continuation.

Activation-based control steers large language models (LLMs) by intervening on their internal representations during inference, and has emerged as an effective paradigm for controlling behaviors such as persona and style. However, existing methods often rely on fixed steering directions or task-specific intervention modules, making them difficult to adapt to fine-grained concepts and compositional constraints. We propose UniSteer, a text-guided activation flow matching model that learns a conditional distribution over residual-stream activations from natural-language conditions. Instead of fitting a separate intervention for each target behavior, UniSteer learns a universal conditional velocity field in activation space. At inference time, UniSteer performs flow inversion by partially transporting a source activation toward a latent state and regenerating it under a target textual condition before injecting it back into the frozen LLM. The same conditional model supports activation-space classification by selecting the textual label with the lowest reconstruction energy. Experiments on three target LLMs show that UniSteer provides a unified interface across behavioral control, truthfulness steering, fine-grained concept steering, multi-constraint instruction following, and activation-space classification.

High-content microscopy enables systematic profiling of cellular responses to chemical perturbations, but the scale of the chemical space makes exhaustive phenotypic characterization experimentally infeasible. This motivates computational models that can predict image-derived phenotypes without acquiring the corresponding treated cells. We formulate molecule-induced phenotype prediction as an inductive conditional transport problem in image representation space. Given a negative-control phenotype and the structure of a molecule, we aim to predict the phenotype induced by the corresponding molecule. We first evaluate classical optimal transport baselines and show that static couplings do not yield useful predictions on large-scale phenotypic image datasets. We then introduce a molecule-conditioned Neural Optimal Transport (NOT) model with a Monge-Gap regularization training objective that learns to transport negative-control unperturbed phenotypes toward perturbed phenotypes using molecular structure as conditioning information. NOT recovers molecule-specific phenotypic effects while reducing microscopy-associated technical variation, thereby facilitating comparisons across experimental batches. On unseen active molecules, the model outperforms baseline approaches, demonstrating that chemically conditioned transport can generalize beyond the molecules observed during training. We identified the molecular encoder as the main limitation to this generalization, while transport in a compressed representation space improves performance and scalability. These results establish NOT as a promising framework for predicting cellular phenotypes from molecular structure and negative-control phenotypes, while highlighting the development of more informative molecular representations as a key direction for improving out-of-distribution performance.

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to couple the base and the target densities. This enables us to incorporate information about class labels or continuous embeddings to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.

Learning the response of single-cells to various treatments offers great potential to enable targeted therapies. In this context, neural optimal transport (OT) has emerged as a principled methodological framework because it inherently accommodates the challenges of unpaired data induced by cell destruction during data acquisition. However, most existing OT approaches are incapable of conditioning on different treatment contexts (e.g., time, drug treatment, drug dosage, or cell type) and we still lack methods that unanimously show promising generalization performance to unseen treatments. Here, we propose the Conditional Monge Gap which learns OT maps conditionally on arbitrary covariates. We demonstrate its value in predicting single-cell perturbation responses conditional to one or multiple drugs, a drug dosage, or combinations thereof. We find that our conditional models achieve results comparable and sometimes even superior to the condition-specific state-of-the-art on scRNA-seq as well as multiplexed protein imaging data. Notably, by aggregating data across conditions we perform cross-task learning which unlocks remarkable generalization abilities to unseen drugs or drug dosages, widely outperforming other conditional models in capturing heterogeneity (i.e., higher moments) in the perturbed population. Finally, by scaling to hundreds of conditions and testing on unseen drugs, we narrow the gap between structure-based and effect-based drug representations, suggesting a promising path to the successful prediction of perturbation effects for unseen treatments.

Flow matching approaches to imaging inverse problems commonly incorporate measurements in two ways. Conditioning-based approaches supply measurement-derived information as a network input, often through concatenation, while inference-guided approaches combine an unconditional velocity field with a separate data-consistency update. In these common formulations, the forward model is not explicitly enforced within the learned conditional velocity field. We propose a principled parametrization of the measurement-conditional velocity field to solve inverse problems. Under linear interpolation, we express the conditional velocity v(x_t,t,y) in terms of the posterior mean E[x_1 | x_t,y], and characterize that mean as the unique minimizer of a variational objective whose data-consistency term is explicit. We further prove that the velocity field defines a probability flow from the source distribution to the measurement-conditioned posterior. Splitting the variational objective yields a conditional velocity parameterization with operator-dependent data-consistency updates, which we train end-to-end under the flow-matching objective, with no additional guidance at inference. Our method achieves state-of-the-art PSNR with 50times fewer function evaluations than the strongest flow baseline. Varying the sampling steps provides test-time control over the distortion-perception trade-off without retraining.

A central challenge in scaling flow models is maintaining stability in training and integration. In flow matching, this stability depends critically on how source samples are paired with data. Independent random pairing introduces substantial stochasticity, motivating minibatch optimal transport (OT) to construct online source-target couplings that reduce this randomness (Tong et al., 2024). However, as applications extend to larger and higher-dimensional datasets, small minibatches can limit these benefits, while scaling the coupling to substantially larger minibatches at every iteration is computationally prohibitive. Moreover, how OT coupling affects the full stochastic gradient and integration remains theoretically unresolved. In this work, we develop a scalable online entropic OT approximation that enables substantially larger source-target couplings while preserving the benefits of OT. We close the theoretical gap for both unregularized and entropic OT by proving strict reductions in full stochastic-gradient variance and integration error relative to independent pairing, and further characterize how these improvements strengthen as the online coupling minibatch grows. Experiments on single-cell gene interpolation-flow and image-flow training validate these predictions, showing that larger online couplings consistently improve training stability and final generation performance.

Optimal transport (OT) aims to find a map T that transports mass from one probability measure to another while minimizing a cost function. Recently, neural OT solvers have gained popularity in high dimensional biological applications such as drug perturbation, due to their superior computational and memory efficiency compared to traditional exact Sinkhorn solvers. However, the overly complex high dimensional maps learned by neural OT solvers often suffer from poor interpretability. Prior work addressed this issue in the context of exact OT solvers by introducing displacement-sparse maps via designed elastic cost, but such method failed to be applied to neural OT settings. In this work, we propose an intuitive and theoretically grounded approach to learning displacement-sparse maps within neural OT solvers. Building on our new formulation, we introduce a novel smoothed ell_0 regularizer that outperforms the ell_1 based alternative from prior work. Leveraging Input Convex Neural Network's flexibility, we further develop a heuristic framework for adaptively controlling sparsity intensity, an approach uniquely enabled by the neural OT paradigm. We demonstrate the necessity of this adaptive framework in large-scale, high-dimensional training, showing not only improved accuracy but also practical ease of use for downstream applications.

We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions pi_0,pi_1 on R^d, of minimizing a transport cost E[c(X_1-X_0)] in the set of couplings (X_0,X_1) whose marginal distributions on X_0,X_1 equals pi_0,pi_1, respectively, where c is a cost function. Our method iteratively constructs a sequence of neural ordinary differentiable equations (ODE), each learned by solving a simple unconstrained regression problem, which monotonically reduce the transport cost while automatically preserving the marginal constraints. This yields a monotonic interior approach that traverses inside the set of valid couplings to decrease the transport cost, which distinguishes itself from most existing approaches that enforce the coupling constraints from the outside. The main idea of the method draws from rectified flow, a recent approach that simultaneously decreases the whole family of transport costs induced by convex functions c (and is hence multi-objective in nature), but is not tailored to minimize a specific transport cost. Our method is a single-object variant of rectified flow that guarantees to solve the OT problem for a fixed, user-specified convex cost function c.

Conditional flow matching (CFM) has emerged as a powerful framework for training continuous normalizing flows due to its computational efficiency and effectiveness. However, standard CFM often produces paths that deviate significantly from straight-line interpolations between prior and target distributions, making generation slower and less accurate due to the need for fine discretization at inference. Recent methods enhance CFM performance by inducing shorter and straighter trajectories but typically rely on computationally expensive mini-batch optimal transport (OT). Drawing insights from entropic optimal transport (EOT), we propose Weighted Conditional Flow Matching (W-CFM), a novel approach that modifies the classical CFM loss by weighting each training pair (x, y) with a Gibbs kernel. We show that this weighting recovers the entropic OT coupling up to some bias in the marginals, and we provide the conditions under which the marginals remain nearly unchanged. Moreover, we establish an equivalence between W-CFM and the minibatch OT method in the large-batch limit, showing how our method overcomes computational and performance bottlenecks linked to batch size. Empirically, we test our method on unconditional generation on various synthetic and real datasets, confirming that W-CFM achieves comparable or superior sample quality, fidelity, and diversity to other alternative baselines while maintaining the computational efficiency of vanilla CFM.

We present the Variational Incompressible Optimal Transport (VIOT) operator, a generative neural operator for amortized incompressible density transport. Given a new source-target density pair, VIOT predicts a divergence-free velocity field and generates the full transport trajectory by feed-forward inference, replacing the hour-scale per-pair optimization used by adjoint fluid solvers and differentiable simulation baselines. The system consists of three components: a stream-function or vector-potential representation that enforces incompressibility by construction, a regularized incompressible transport objective that balances endpoint accuracy and flow smoothness, and a Fourier Neural Operator backbone that amortizes the solve across new pairs and grid resolutions. Together, these components make incompressible transport a reusable neural operator that facilitates various transport processes. Further, the generative capability extends beyond the training distribution, with VIOT producing incompressible transports for user-drawn source-target pairs in a real-time interactive system. We demonstrate VIOT on 2D and 3D density-transport benchmarks. Both 2D and 3D rollouts complete in seconds per pair, while per-instance baselines in our 2D comparisons optimize each new pair from scratch and require on the order of an hour, a roughly 10^4times online speedup.

Predicting single-cell perturbation responses requires mapping between two unpaired populations of cells measured before and after the perturbation. Optimal transport (OT) provides a principled framework for constructing such mappings by minimizing the transport cost, and neural OT methods (e.g., CellOT) have been employed for this prediction task. However, these methods rely on a fixed transport cost, typically the squared Euclidean distance. This cost may connect cells with the wrong responses. Lineage tracing provides the information needed to learn the cost, as heritable barcodes link the cells of the same clone before and after the perturbation. Here, we propose \CloneOT, a neural inverse OT method that learns both the transport cost and the transport map from partially lineage-traced single-cell datasets. \CloneOT\ represents the cost as a neural network. It learns the cost by minimizing the extra cost that the lineage adds to the optimal transport, so that the cheapest transport keeps the cells of each clone together. Our experiments demonstrate that \CloneOT\ recovers the correct transport on 2D datasets, while the squared Euclidean cost fails. Moreover, \CloneOT\ outperforms five baselines on 14 lineage-tracing perturbation scenarios of cancer cells, for both seen and unseen clones. Furthermore, we show that \CloneOT\ reveals the clone-specific transcriptional programs of drug-tolerant persister cells and forecasts the fate of each clone from its cells before treatment. Our implementation is available at https://github.com/poseidonchan/CloneOT.

Adaptation methods have been a workhorse for unlocking the transformative power of pre-trained diffusion models in diverse applications. Existing approaches often abstract adaptation objectives as a reward function and steer diffusion models to generate high-reward samples. However, these approaches can incur high computational overhead due to additional training, or rely on stringent assumptions on the reward such as differentiability. Moreover, despite their empirical success, theoretical justification and guarantees are seldom established. In this paper, we propose DOIT (Doob-Oriented Inference-time Transformation), a training-free and computationally efficient adaptation method that applies to generic, non-differentiable rewards. The key framework underlying our method is a measure transport formulation that seeks to transport the pre-trained generative distribution to a high-reward target distribution. We leverage Doob's h-transform to realize this transport, which induces a dynamic correction to the diffusion sampling process and enables efficient simulation-based computation without modifying the pre-trained model. Theoretically, we establish a high probability convergence guarantee to the target high-reward distribution via characterizing the approximation error in the dynamic Doob's correction. Empirically, on D4RL offline RL benchmarks, our method consistently outperforms state-of-the-art baselines while preserving sampling efficiency.

Recent successful generative models are trained by fitting a neural network to an a-priori defined tractable probability density path taking noise to training examples. In this paper we investigate the space of Gaussian probability paths, which includes diffusion paths as an instance, and look for an optimal member in some useful sense. In particular, minimizing the Kinetic Energy (KE) of a path is known to make particles' trajectories simple, hence easier to sample, and empirically improve performance in terms of likelihood of unseen data and sample generation quality. We investigate Kinetic Optimal (KO) Gaussian paths and offer the following observations: (i) We show the KE takes a simplified form on the space of Gaussian paths, where the data is incorporated only through a single, one dimensional scalar function, called the data separation function. (ii) We characterize the KO solutions with a one dimensional ODE. (iii) We approximate data-dependent KO paths by approximating the data separation function and minimizing the KE. (iv) We prove that the data separation function converges to 1 in the general case of arbitrary normalized dataset consisting of n samples in d dimension as n/drightarrow 0. A consequence of this result is that the Conditional Optimal Transport (Cond-OT) path becomes kinetic optimal as n/drightarrow 0. We further support this theory with empirical experiments on ImageNet.

Optimal transport aligns samples across distributions by minimizing the transportation cost between them, e.g., the geometric distances. Yet, it ignores coherence structure in the data such as clusters, does not handle outliers well, and cannot integrate new data points. To address these drawbacks, we propose InfoOT, an information-theoretic extension of optimal transport that maximizes the mutual information between domains while minimizing geometric distances. The resulting objective can still be formulated as a (generalized) optimal transport problem, and can be efficiently solved by projected gradient descent. This formulation yields a new projection method that is robust to outliers and generalizes to unseen samples. Empirically, InfoOT improves the quality of alignments across benchmarks in domain adaptation, cross-domain retrieval, and single-cell alignment.

In optimal transport (OT), a Monge map is known as a mapping that transports a source distribution to a target distribution in the most cost-efficient way. Recently, multiple neural estimators for Monge maps have been developed and applied in diverse unpaired domain translation tasks, e.g. in single-cell biology and computer vision. However, the classic OT framework enforces mass conservation, which makes it prone to outliers and limits its applicability in real-world scenarios. The latter can be particularly harmful in OT domain translation tasks, where the relative position of a sample within a distribution is explicitly taken into account. While unbalanced OT tackles this challenge in the discrete setting, its integration into neural Monge map estimators has received limited attention. We propose a theoretically grounded method to incorporate unbalancedness into any Monge map estimator. We improve existing estimators to model cell trajectories over time and to predict cellular responses to perturbations. Moreover, our approach seamlessly integrates with the OT flow matching (OT-FM) framework. While we show that OT-FM performs competitively in image translation, we further improve performance by incorporating unbalancedness (UOT-FM), which better preserves relevant features. We hence establish UOT-FM as a principled method for unpaired image translation.

Entropic optimal transport (EOT) in continuous spaces with quadratic cost is a classical tool for solving the domain translation problem. In practice, recent approaches optimize a weak dual EOT objective depending on a single potential, but doing so is computationally not efficient due to the intractable log-partition term. Existing methods typically resolve this obstacle in one of two ways: by significantly restricting the transport family to obtain closed-form normalization (via Gaussian-mixture parameterizations), or by using general neural parameterizations that require simulation-based training procedures. We propose Variational Entropic Optimal Transport (VarEOT), based on an exact variational reformulation of the log-partition log E[exp(cdot)] as a tractable minimization over an auxiliary log-normalizer. This yields a differentiable learning objective optimized with stochastic gradients and avoids the necessity of MCMC simulations during the training. We provide theoretical guarantees, including finite-sample generalization bounds and approximation results under universal function approximation. Experiments on synthetic data and unpaired image-to-image translation demonstrate competitive or improved translation quality, while comparisons within the solvers that use the same weak dual EOT objective support the benefit of the proposed optimization principle. The code for our solver can be found at https://github.com/DrEternity/VarEOT .

Energy-based models (EBMs) are known in the Machine Learning community for decades. Since the seminal works devoted to EBMs dating back to the noughties, there have been a lot of efficient methods which solve the generative modelling problem by means of energy potentials (unnormalized likelihood functions). In contrast, the realm of Optimal Transport (OT) and, in particular, neural OT solvers is much less explored and limited by few recent works (excluding WGAN-based approaches which utilize OT as a loss function and do not model OT maps themselves). In our work, we bridge the gap between EBMs and Entropy-regularized OT. We present a novel methodology which allows utilizing the recent developments and technical improvements of the former in order to enrich the latter. From the theoretical perspective, we prove generalization bounds for our technique. In practice, we validate its applicability in toy 2D and image domains. To showcase the scalability, we empower our method with a pre-trained StyleGAN and apply it to high-res AFHQ 512times 512 unpaired I2I translation. For simplicity, we choose simple short- and long-run EBMs as a backbone of our Energy-guided Entropic OT approach, leaving the application of more sophisticated EBMs for future research. Our code is available at: https://github.com/PetrMokrov/Energy-guided-Entropic-OT

Test-time adaptation (TTA) typically assumes that model parameters can be updated at inference time. This assumption is restrictive for inference-only accelerators, frozen or third-party models, and memory-constrained deployments, and standard BatchNorm-based TTA configurations may also become inactive on architectures without BatchNorm. We study adaptation when the learned model must remain frozen. We introduce CASTER, a gradient-free method that stores source class statistics in a discriminative subspace, estimates a class-shared affine transformation from target-batch moments, and analytically transports the source class distributions before classification. CASTER requires no backward pass, optimizer state, or stored source feature bank. Across four backbones and seven datasets, it outperforms k-NN on identical frozen features in 27 of 28 backbone-dataset settings while retaining a median of 18x less state. Affine transport is not always reliable. On ImageNet-C, where batches contain only 64 samples for 1000 classes, unconditional transport loses 21.2 top-1 points. We therefore introduce an empirical residual-to-margin transportability certificate. Across 307 evaluation cells, every transport losing more than 10 points has certificate value above 3.9, although benign and destructive regimes are not perfectly separated. Gating converts an average -3.35-point effect of unconditional transport into a +1.69-point gain, and performance remains within 0.3 points of the best threshold over a broad threshold range. Finally, we show that this certificate is mechanism-specific: when applied to Tent, it accepts only 4.3% of updates and preserves 0.6% of Tent's available gain. These results position CASTER as a lightweight adaptation mechanism for frozen-model deployment, together with an explicit account of when its safety signal is informative and when it is not.

Robust generalization under distribution shift remains a key challenge for conditional generative modeling: conditional flow-based methods often fit the training conditions well but fail to extrapolate to unseen ones. We introduce SP-FM, a shortest-path flow-matching framework that improves out-of-distribution (OOD) generalization by conditioning both the base distribution and the flow field on the condition. Specifically, SP-FM learns a condition-dependent base distribution parameterized as a flexible, learnable mixture, together with a condition-dependent vector field trained via shortest-path flow matching. Conditioning the base allows the model to adapt its starting distribution across conditions, enabling smooth interpolation and more reliable extrapolation beyond the observed training range. We provide theoretical insights into the resulting conditional transport and show how mixture-conditioned bases enhance robustness under shift. Empirically, SP-FM is effective across heterogeneous domains, including predicting responses to unseen perturbations in single-cell transcriptomics and modeling treatment effects in high-content microscopy--based drug screening. Overall, SP-FM provides a simple yet effective plug-in strategy for improving conditional generative modeling and OOD generalization across diverse domains.

We propose a novel neural algorithm for the fundamental problem of computing the entropic optimal transport (EOT) plan between continuous probability distributions which are accessible by samples. Our algorithm is based on the saddle point reformulation of the dynamic version of EOT which is known as the Schr\"odinger Bridge problem. In contrast to the prior methods for large-scale EOT, our algorithm is end-to-end and consists of a single learning step, has fast inference procedure, and allows handling small values of the entropy regularization coefficient which is of particular importance in some applied problems. Empirically, we show the performance of the method on several large-scale EOT tasks. https://github.com/ngushchin/EntropicNeuralOptimalTransport

One-step generative models construct a static generator through iterative training-time transport. Existing transport objectives primarily assess distributional motion, although a neural generator needs to realize the requested sample displacements jointly through shared parameter updates. The training-time construction raises the question: once training becomes the iterative process that constructs the final one-step map, what to optimize: the next distributional move, or the route by which the finite generator learns the final map? To address the question, we introduce Training Dynamics Action (TDAction), which selects transport targets according to local shared-parameter realization cost while retaining a prescribed level of distributional progress. We formulate the cost as a soft-terminal control problem and derive a closed-form Batch Tangent Action-to-Go value that accounts for parameter effort and terminal mismatch. The criterion captures cross-sample interactions omitted by independent pairwise costs; under isotropic mobility, the criterion agrees with quadratic Euclidean assignment for deterministic balanced couplings. Randomized tangent probes provide a low-rank implementation that constructs shared detached targets without adding an inference-time trajectory. Controlled studies examine the relationship between generator geometry, transport selection, and realized local action. On ImageNet 256times256, TDAction attains an FID below 1.1 without distillation.

Inference-time alignment of pretrained text-to-image models is typically performed along a single control axis, such as classifier-free guidance, attention editing, or reward-based latent perturbations. This limitation prevents modeling joint dependencies between conditioning and latent variables and hinders transfer across generative transports. We propose PG-MAP, a training-free framework that formulates inference-time alignment as a trajectory-level Gibbs-MAP / proximal energy optimization over the conditioning c and latent state z_t via a forward-consistency coupling, optionally guided by a frozen preference reward. This joint formulation enables coordinated updates across modalities while remaining compatible with both diffusion and flow-matching models through transport-specific adaptations. Across diffusion backbones (SD~1.5, SDXL), PG-MAP consistently improves alignment metrics such as PickScore and Aesthetic, and can be effectively combined with tuned classifier-free guidance to achieve the strongest overall performance. On flow-matching models (SD3.5-medium), the framework reduces to a latent-only variant, achieving 91.9% PickScore and 75.7% HPS win rates against a static baseline, with controlled experiments ruling out noise-related artifacts. Human evaluations further confirm consistent preference over strong baselines, including tuned CFG and compute-matched universal guidance. Finally, an oracle-routing analysis shows that the relative importance of conditioning and latent optimization depends on prompt types, surfacing further headroom that a per-prompt selector could exploit.

We present Free energy Estimators with Adaptive Transport (FEAT), a novel framework for free energy estimation -- a critical challenge across scientific domains. FEAT leverages learned transports implemented via stochastic interpolants and provides consistent, minimum-variance estimators based on escorted Jarzynski equality and controlled Crooks theorem, alongside variational upper and lower bounds on free energy differences. Unifying equilibrium and non-equilibrium methods under a single theoretical framework, FEAT establishes a principled foundation for neural free energy calculations. Experimental validation on toy examples, molecular simulations, and quantum field theory demonstrates improvements over existing learning-based methods.

We study behavior-regularized reinforcement learning (RL), where regularization toward a reference distribution (the dataset in offline RL or the base model in LLM RL finetuning) is essential to prevent value over-optimization caused by erroneous out-of-distribution extrapolation. Existing methods either rely on reparameterized policy gradient, which are difficult to scale to large generative models, or on reject sampling, which can be overly conservative when attempting to move beyond the behavior support. In this paper, we propose Value Gradient Flow (VGF), a scalable new paradigm for behavior-regularized RL. VGF casts behavior-regularized RL as an optimal transport problem that maps the reference distribution to the value-induced optimal policy distribution. We solve this transport problem via discrete gradient flow, where value gradients guide particles initialized from the reference distribution. Our analysis shows that VGF imposes regularization implicitly by controlling the transport budget. VGF eliminates explicit policy parameterization while remaining expressive and flexible, this enables adaptive test-time scaling by adjusting the transport budget. Extensive experiments demonstrate that VGF significantly outperforms prior methods, achieving state-of-the-art results on offline RL benchmarks (D4RL, OGBench) and LLM RL tasks. Code and runs can be found at https://ryanxhr.github.io/vgf.

We consider the problem of estimating the optimal transport map between two probability distributions, P and Q in mathbb R^d, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points in mathbb R^d. We study a computationally efficient estimator initially proposed by Pooladian and Niles-Weed (2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate n^{-1/2}, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.

Regularized optimal transport (OT) is now increasingly used as a loss or as a matching layer in neural networks. Entropy-regularized OT can be computed using the Sinkhorn algorithm but it leads to fully-dense transportation plans, meaning that all sources are (fractionally) matched with all targets. To address this issue, several works have investigated quadratic regularization instead. This regularization preserves sparsity and leads to unconstrained and smooth (semi) dual objectives, that can be solved with off-the-shelf gradient methods. Unfortunately, quadratic regularization does not give direct control over the cardinality (number of nonzeros) of the transportation plan. We propose in this paper a new approach for OT with explicit cardinality constraints on the transportation plan. Our work is motivated by an application to sparse mixture of experts, where OT can be used to match input tokens such as image patches with expert models such as neural networks. Cardinality constraints ensure that at most k tokens are matched with an expert, which is crucial for computational performance reasons. Despite the nonconvexity of cardinality constraints, we show that the corresponding (semi) dual problems are tractable and can be solved with first-order gradient methods. Our method can be thought as a middle ground between unregularized OT (recovered in the limit case k=1) and quadratically-regularized OT (recovered when k is large enough). The smoothness of the objectives increases as k increases, giving rise to a trade-off between convergence speed and sparsity of the optimal plan.

Continuous normalizing flows (CNFs) are an attractive generative modeling technique, but they have been held back by limitations in their simulation-based maximum likelihood training. We introduce the generalized conditional flow matching (CFM) technique, a family of simulation-free training objectives for CNFs. CFM features a stable regression objective like that used to train the stochastic flow in diffusion models but enjoys the efficient inference of deterministic flow models. In contrast to both diffusion models and prior CNF training algorithms, CFM does not require the source distribution to be Gaussian or require evaluation of its density. A variant of our objective is optimal transport CFM (OT-CFM), which creates simpler flows that are more stable to train and lead to faster inference, as evaluated in our experiments. Furthermore, we show that when the true OT plan is available, our OT-CFM method approximates dynamic OT. Training CNFs with CFM improves results on a variety of conditional and unconditional generation tasks, such as inferring single cell dynamics, unsupervised image translation, and Schr\"odinger bridge inference.

We present an online reinforcement learning (RL) algorithm for fine-tuning flow-matching policies in continuous-control problems. Our key insight is to view RL-based policy improvement as a transport of action densities towards regions of high reward, which naturally aligns with the transport formulation of flow matching models. Prior methods either approximate the current or optimal policy distribution or resort to distillation, which introduces biased gradients or sacrifices multimodal modeling capacity. In contrast, our approach for RL with Density Transport, which we name RLDT, constructs a transport field from a maximum-entropy RL objective using Stein Variational Gradient Descent (SVGD). Then, it finetunes a pretrained flow matching policy to align with this field. Training with this alignment objective is nontrivial because flow-matching policies generate actions via a multi-step process, making direct gradient-based optimization challenging. To overcome this challenge and stabilize training, we approximate policy actions from intermediate denoising steps via expected-target estimation. This allows the transport-field update to propagate into the network parameters without unstable backpropagation through time. Experimental results demonstrate that RLDT outperforms competitive baselines in reward quality and convergence speed. This performance holds across diverse continuous-control tasks, encompassing both dense and sparse rewards, as well as state- and vision-based long-horizon robot manipulation. The project webpage is https://rpfey.github.io/rldt/{https://rpfey.github.io/rldt/}.

Diffusion and flow matching models generate samples by learning time-dependent vector fields whose integration transports noise to data, requiring tens to hundreds of network evaluations at inference. We instead learn the transport map directly. We propose Optimal Transport Neural Flow Matching (OT-NFM), an ODE-free generative framework that parameterizes the flow map with neural flows, enabling true one-step generation with a single forward pass. We show that naive flow-map training suffers from mean collapse, where inconsistent noise-data pairings drive all outputs toward the data mean. We prove that consistent coupling is necessary for non-degenerate learning and address this using optimal transport pairings with scalable minibatch and online coupling strategies. Experiments on synthetic benchmarks and image generation tasks (MNIST and CIFAR-10) demonstrate competitive sample quality while reducing inference to a single network evaluation.

One-step generators enable high-quality visual generation with a single network evaluation, but their post-training is difficult: general implicit generators provide neither tractable likelihoods nor denoising trajectories, and many rewards are non-differentiable. We introduce Reward-Weighted Transport Distillation (RWTD), a post-training method that requires only generated samples and scalar reward evaluations. Rather than aligning solely to the conventional reward-tilted reference distribution, RWTD constructs an adaptive target that mixes separately tilted current and reference distributions. The current component incorporates improvements discovered during training, while the reference component anchors the target to the pretrained generator. RWTD realizes this target through feature-space optimal transport and fixed-point regression. Theoretical analysis shows that the fixed-point distributions of RWTD interpolate between off-policy reward tilting of the reference and on-policy tilting of the current model, providing a principled approach to balancing reward adaptation with retention of prior knowledge. Empirically, RWTD substantially improves the GenEval score of the one-step SANA Sprint 1.6B backbone from 0.73 to 0.80, while separate preference alignment experiments demonstrate strong cross-reward generalization that yields balanced improvements and preservation of compositional capabilities.

Recent studies have explored autoregressive models for image generation, with promising results, and have combined diffusion models with autoregressive frameworks to optimize image generation via diffusion losses. In this study, we present a theoretical analysis of diffusion and autoregressive models with diffusion loss, highlighting the latter's advantages. We present a theoretical comparison of conditional diffusion and autoregressive diffusion with diffusion loss, demonstrating that patch denoising optimization in autoregressive models effectively mitigates condition errors and leads to a stable condition distribution. Our analysis also reveals that autoregressive condition generation refines the condition, causing the condition error influence to decay exponentially. In addition, we introduce a novel condition refinement approach based on Optimal Transport (OT) theory to address ``condition inconsistency''. We theoretically demonstrate that formulating condition refinement as a Wasserstein Gradient Flow ensures convergence toward the ideal condition distribution, effectively mitigating condition inconsistency. Experiments demonstrate the superiority of our method over diffusion and autoregressive models with diffusion loss methods.

Reconstructing continuous speech from scalp electroencephalography (EEG) remains fundamentally challenging. EEG provides a weak, spatially diffuse, and highly variable measurement of distributed cortical activity, whereas speech is organized as a coherent acoustic trajectory with strong harmonic and temporal structure. The resulting mismatch makes waveform regression unstable and causes stochastic multi-step generation to be sensitive to artifact-dependent conditioning and subject variability. We introduce NeuroSonic, a conditional flow-matching framework for EEG-to-speech reconstruction. Instead of predicting waveforms directly or refining them through stochastic denoising, NeuroSonic learns a deterministic probability-flow velocity field that transports a noise-corrupted acoustic state toward clean speech under EEG conditioning. EEG and audio are embedded into a shared token space and processed by a time-conditioned gated Transformer that parameterizes the transport ordinary differential equation. This formulation models trajectory evolution explicitly while avoiding iterative stochastic sampling. We evaluate NeuroSonic on the CineBrain and EAV benchmarks under cross-subject evaluation. Across both datasets, the proposed method improves distributional realism, spectral fidelity, and perceptual quality over representative GAN-, diffusion-, and mean-flow baselines, with up to a 26.3\% gain in overall perceptual quality. The performance gap is most evident in artifact-heavy segments, where conditioning variability is strongest. These findings indicate that deterministic conditional transport provides a stable and effective formulation for EEG-driven speech reconstruction. Code is available at https://github.com/Y-Research-SBU/NeuroSonic/ .

Reward fine-tuning aims to update a pre-trained flow-based generative model to improve the downstream reward of its generated samples. Existing methods typically formulate this problem as sampling from a reward-tilted distribution, the solution to a KL-regularized reward-maximization problem. Here, we introduce an optimal transport regularizer built directly from the pre-trained drift. Unlike KL reward tilting, the resulting objective transports individual samples toward higher reward rather than reweighting the base distribution. We show that the resulting problem is equivalent to a deterministic optimal control problem on the flow. Given a pre-trained flow map, this equivalence yields a simulation-free reinforcement learning algorithm for fine-tuning generative flows. We call the resulting framework Wasserstein-Tilted Flow Maps (WTF), the first end-to-end fine-tuning recipe native to flow maps. The output is a fine-tuned flow map that retains strong reward-aligned performance at few-step inference budgets without post-hoc distillation. Experiments on ImageNet-256 and text-to-image show that WTF achieves higher reward with comparable or higher diversity than baselines, while requiring up to 280times less training compute. More broadly, we argue that accelerated samplers such as flow maps are essential infrastructure for efficient post-training, and that the dominant KL-regularized formulation is only one of many choices worth revisiting.

Optimal transport (OT) theory focuses, among all maps T:R^drightarrow R^d that can morph a probability measure onto another, on those that are the ``thriftiest'', i.e. such that the averaged cost c(x, T(x)) between x and its image T(x) be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when c is the ell_2^2 distance, e.g., using entropic maps [Pooladian'22], or neural networks [Makkuva'20, Korotin'20]. We propose a new model for transport maps, built on a family of translation invariant costs c(x, y):=h(x-y), where h:=1{2}|cdot|_2^2+tau and tau is a regularizer. We propose a generalization of the entropic map suitable for h, and highlight a surprising link tying it with the Bregman centroids of the divergence D_h generated by h, and the proximal operator of tau. We show that choosing a sparsity-inducing norm for tau results in maps that apply Occam's razor to transport, in the sense that the displacement vectors Delta(x):= T(x)-x they induce are sparse, with a sparsity pattern that varies depending on x. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data, in the 34000-d space of gene counts for cells, without using dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.

Transfer learning has become a central paradigm in modern machine learning, yet it suffers from the long-standing problem of negative transfer, where leveraging source representations can harm rather than help performance on the target task. Although empirical remedies have been proposed, there remains little theoretical understanding of how to reliably avoid negative transfer. In this paper, we investigate a simple yet remarkably effective strategy: augmenting frozen, pretrained source-side features with a trainable target-side encoder that adapts target features to capture residual signals overlooked by models pretrained on the source data. We show this residual feature integration strategy is sufficient to provably prevent negative transfer, by establishing theoretical guarantees that it has no worse convergence rate than training from scratch under the informative class of target distributions up to logarithmic factors, and that the convergence rate can transition seamlessly from nonparametric to near-parametric when source representations are informative. To our knowledge, this is the first theoretical work that ensures protection against negative transfer. We carry out extensive numerical experiments across image, text and tabular benchmarks, and empirically verify that the method consistently safeguards performance under distribution shift, label noise, semantic perturbation, and class imbalance. We additionally demonstrate that this residual integration mechanism uniquely supports adapt-time multimodality extension, enabling a pretrained single-cell foundation model to incorporate spatial signals for lymph-node anatomical classification despite the source model being trained without them. Our study thus advances the theory of safe transfer learning, and provides a principled approach that is simple, robust, architecture-agnostic, and broadly applicable.

We introduce a new paradigm for generative modeling built on Continuous Normalizing Flows (CNFs), allowing us to train CNFs at unprecedented scale. Specifically, we present the notion of Flow Matching (FM), a simulation-free approach for training CNFs based on regressing vector fields of fixed conditional probability paths. Flow Matching is compatible with a general family of Gaussian probability paths for transforming between noise and data samples -- which subsumes existing diffusion paths as specific instances. Interestingly, we find that employing FM with diffusion paths results in a more robust and stable alternative for training diffusion models. Furthermore, Flow Matching opens the door to training CNFs with other, non-diffusion probability paths. An instance of particular interest is using Optimal Transport (OT) displacement interpolation to define the conditional probability paths. These paths are more efficient than diffusion paths, provide faster training and sampling, and result in better generalization. Training CNFs using Flow Matching on ImageNet leads to consistently better performance than alternative diffusion-based methods in terms of both likelihood and sample quality, and allows fast and reliable sample generation using off-the-shelf numerical ODE solvers.

Flow matching has recently emerged as a promising alternative to diffusion-based generative models, particularly for text-to-image generation. Despite its flexibility in allowing arbitrary source distributions, most existing approaches rely on a standard Gaussian distribution, a choice inherited from diffusion models, and rarely consider the source distribution itself as an optimization target in such settings. In this work, we show that principled design of the source distribution is not only feasible but also beneficial at the scale of modern text-to-image systems. Specifically, we propose learning a condition-dependent source distribution under flow matching objective that better exploit rich conditioning signals. We identify key failure modes that arise when directly incorporating conditioning into the source, including distributional collapse and instability, and show that appropriate variance regularization and directional alignment between source and target are critical for stable and effective learning. We further analyze how the choice of target representation space impacts flow matching with structured sources, revealing regimes in which such designs are most effective. Extensive experiments across multiple text-to-image benchmarks demonstrate consistent and robust improvements, including up to a 3x faster convergence in FID, highlighting the practical benefits of a principled source distribution design for conditional flow matching.

Diffusion models are commonly interpreted as learning the score function, i.e., the gradient of the log-density of noisy data. However, this assumption implies that the target of learning is a conservative vector field, which is not enforced by the neural network architectures used in practice. We present numerical evidence that trained diffusion networks violate both integral and differential constraints required of true score functions, demonstrating that the learned vector fields are not conservative. Despite this, the models perform remarkably well as generative mechanisms. To explain this apparent paradox, we advocate a new theoretical perspective: diffusion training is better understood as flow matching to the velocity field of a Wasserstein Gradient Flow (WGF), rather than as score learning for a reverse-time stochastic differential equation. Under this view, the "probability flow" arises naturally from the WGF framework, eliminating the need to invoke reverse-time SDE theory and clarifying why generative sampling remains successful even when the neural vector field is not a true score. We further show that non-conservative errors from neural approximation do not necessarily harm density transport. Our results advocate for adopting the WGF perspective as a principled, elegant, and theoretically grounded framework for understanding diffusion generative models.

Optimal transport (OT) compares probability distributions by computing a meaningful alignment between their samples. CO-optimal transport (COOT) takes this comparison further by inferring an alignment between features as well. While this approach leads to better alignments and generalizes both OT and Gromov-Wasserstein distances, we provide a theoretical result showing that it is sensitive to outliers that are omnipresent in real-world data. This prompts us to propose unbalanced COOT for which we provably show its robustness to noise in the compared datasets. To the best of our knowledge, this is the first such result for OT methods in incomparable spaces. With this result in hand, we provide empirical evidence of this robustness for the challenging tasks of heterogeneous domain adaptation with and without varying proportions of classes and simultaneous alignment of samples and features across single-cell measurements.

Randomized controlled trials often do not represent the populations where decisions are made, and covariate shift across studies can invalidate standard IPD meta-analysis and transport estimators. We propose a placebo-anchored transport framework that treats source-trial outcomes as abundant proxy signals and target-trial placebo outcomes as scarce, high-fidelity gold labels to calibrate baseline risk. A low-complexity (sparse) correction anchors proxy outcome models to the target population, and the anchored models are embedded in a cross-fitted doubly robust learner, yielding a Neyman-orthogonal, target-site doubly robust estimator for patient-level heterogeneous treatment effects when target treated outcomes are available. We distinguish two regimes: in connected targets (with a treated arm), the method yields target-identified effect estimates; in disconnected targets (placebo-only), it reduces to a principled screen--then--transport procedure under explicit working-model transport assumptions. Experiments on synthetic data and a semi-synthetic IHDP benchmark evaluate pointwise CATE accuracy, ATE error, ranking quality for targeting, decision-theoretic policy regret, and calibration. Across connected settings, the proposed method is best or near-best and improves substantially over proxy-only, target-only, and transport baselines at small target sample sizes; in disconnected settings, it retains strong ranking performance for targeting while pointwise accuracy depends on the strength of the working transport condition.

The Entropic Optimal Transport (EOT) problem and its dynamic counterpart, the Schrödinger bridge (SB) problem, play an important role in modern machine learning, linking generative modeling with optimal transport theory. While recent advances in discrete diffusion and flow models have sparked growing interest in applying SB methods to discrete domains, there is still no reliable way to evaluate how well these methods actually solve the underlying problem. We address this challenge by introducing a benchmark for SB on discrete spaces. Our construction yields pairs of probability distributions with analytically known SB solutions, enabling rigorous evaluation. As a byproduct of building this benchmark, we obtain two new SB algorithms, DLightSB and DLightSB-M, and additionally extend prior related work to construct the α-CSBM algorithm. We demonstrate the utility of our benchmark by evaluating both existing and new solvers in high-dimensional discrete settings. This work provides the first step toward proper evaluation of SB methods on discrete spaces, paving the way for more reproducible future studies.

Learning the continuous dynamics of a system from snapshots of its temporal marginals is a problem which appears throughout natural sciences and machine learning, including in quantum systems, single-cell biological data, and generative modeling. In these settings, we assume access to cross-sectional samples that are uncorrelated over time, rather than full trajectories of samples. In order to better understand the systems under observation, we would like to learn a model of the underlying process that allows us to propagate samples in time and thereby simulate entire individual trajectories. In this work, we propose Action Matching, a method for learning a rich family of dynamics using only independent samples from its time evolution. We derive a tractable training objective, which does not rely on explicit assumptions about the underlying dynamics and does not require back-propagation through differential equations or optimal transport solvers. Inspired by connections with optimal transport, we derive extensions of Action Matching to learn stochastic differential equations and dynamics involving creation and destruction of probability mass. Finally, we showcase applications of Action Matching by achieving competitive performance in a diverse set of experiments from biology, physics, and generative modeling.

Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.

Survival prediction using whole slide images (WSIs) can be formulated as a multiple instance learning (MIL) problem. However, existing MIL methods often fail to explicitly capture pathological heterogeneity within WSIs, both globally -- through long-tailed morphological distributions, and locally through -- tile-level prediction uncertainty. Optimal transport (OT) provides a principled way of modeling such heterogeneity by incorporating marginal distribution constraints. Building on this insight, we propose OTSurv, a novel MIL framework from an optimal transport perspective. Specifically, OTSurv formulates survival predictions as a heterogeneity-aware OT problem with two constraints: (1) global long-tail constraint that models prior morphological distributions to avert both mode collapse and excessive uniformity by regulating transport mass allocation, and (2) local uncertainty-aware constraint that prioritizes high-confidence patches while suppressing noise by progressively raising the total transport mass. We then recast the initial OT problem, augmented by these constraints, into an unbalanced OT formulation that can be solved with an efficient, hardware-friendly matrix scaling algorithm. Empirically, OTSurv sets new state-of-the-art results across six popular benchmarks, achieving an absolute 3.6% improvement in average C-index. In addition, OTSurv achieves statistical significance in log-rank tests and offers high interpretability, making it a powerful tool for survival prediction in digital pathology. Our codes are available at https://github.com/Y-Research-SBU/OTSurv.

Continuous normalizing flows (CNFs) construct invertible mappings between an arbitrary complex distribution and an isotropic Gaussian distribution using Neural Ordinary Differential Equations (neural ODEs). It has not been tractable on large datasets due to the incremental complexity of the neural ODE training. Optimal Transport theory has been applied to regularize the dynamics of the ODE to speed up training in recent works. In this paper, a temporal optimization is proposed by optimizing the evolutionary time for forward propagation of the neural ODE training. In this appoach, we optimize the network weights of the CNF alternately with evolutionary time by coordinate descent. Further with temporal regularization, stability of the evolution is ensured. This approach can be used in conjunction with the original regularization approach. We have experimentally demonstrated that the proposed approach can significantly accelerate training without sacrifying performance over baseline models.

Autoregressive learning of time-stepping operators provides an effective approach to data-driven partial differential equation (PDE) simulation, yet for conservation laws, they face a fundamental challenge: learned updates may violate global conservation over long rollouts. For the important subclass of mass-conservation-type equations, the problem is compounded by inherent physical bounds (e.g., nonnegativity or concentrations in [0,1]) whose violation further destabilizes predictions. We introduce FluxNet, which learns cumulative transport amounts representing the total conserved quantity redistributed between each cell and a configurable neighborhood over the full surrogate interval. A conservative update guarantees exact discrete conservation by construction; modular capacity-constrained transport heads (L, U, and D) enforce lower bounds, upper bounds, or near-zero dual-bound violations through architectural design. Unlike flux-rate surrogates that require temporal integration and thus inherit CFL constraints, FluxNet involves no such integration; configurable transport neighborhoods enable large-timestep prediction at full spatial resolution. Ghost cells extend the framework to non-periodic boundaries. Experiments on four benchmarks (1D convection--diffusion, 2D shallow water, 1D traffic flow, 2D Cahn--Hilliard) demonstrate exact conservation, structural bound preservation, architecture modularity, and superior stability over flux-rate surrogates at large temporal strides. The code is publicly available at: https://github.com/Lan-zs/FluxNet.

Mini-batch optimal transport (m-OT) has been widely used recently to deal with the memory issue of OT in large-scale applications. Despite their practicality, m-OT suffers from misspecified mappings, namely, mappings that are optimal on the mini-batch level but are partially wrong in the comparison with the optimal transportation plan between the original measures. Motivated by the misspecified mappings issue, we propose a novel mini-batch method by using partial optimal transport (POT) between mini-batch empirical measures, which we refer to as mini-batch partial optimal transport (m-POT). Leveraging the insight from the partial transportation, we explain the source of misspecified mappings from the m-OT and motivate why limiting the amount of transported masses among mini-batches via POT can alleviate the incorrect mappings. Finally, we carry out extensive experiments on various applications such as deep domain adaptation, partial domain adaptation, deep generative model, color transfer, and gradient flow to demonstrate the favorable performance of m-POT compared to current mini-batch methods.

We develop a model-free policy gradient method for discrete-time mean-field control (MFC). In MFC, the policy affects the objective both through the controlled dynamics and through the population distribution. Standard REINFORCE estimators capture the first effect but not the second. We introduce Transport REINFORCE, a transport map-based approach that perturbs a suitable transformation of the population distribution to estimate this missing mean-field contribution. The method applies to both finite and continuous state spaces. In finite state spaces, we perturb the population distribution directly on the probability simplex through a convex combination of the current population weights and random weights. In continuous state spaces, we project the population distribution onto the manifold of Gaussian mixtures, and then randomize it via a transport map that ensures the perturbed law remains within this manifold. We prove consistency of the perturbed objective and gradient as the perturbation vanishes, and derive bias and mean-square error bounds for the resulting sample-based gradient estimator. Numerical experiments on several MFC benchmarks show that Transport REINFORCE improves over standard REINFORCE.

Optimal Transport is a useful metric to compare probability distributions and to compute a pairing given a ground cost. Its entropic regularization variant (eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings. This work focuses on Inverse Optimal Transport (iOT), the problem of inferring the ground cost from samples drawn from a coupling that solves an eOT problem. It is a relevant problem that can be used to infer unobserved/missing links, and to obtain meaningful information about the structure of the ground cost yielding the pairing. On one side, iOT benefits from convexity, but on the other side, being ill-posed, it requires regularization to handle the sampling noise. This work presents an in-depth theoretical study of the l1 regularization to model for instance Euclidean costs with sparse interactions between features. Specifically, we derive a sufficient condition for the robust recovery of the sparsity of the ground cost that can be seen as a far reaching generalization of the Lasso's celebrated Irrepresentability Condition. To provide additional insight into this condition, we work out in detail the Gaussian case. We show that as the entropic penalty varies, the iOT problem interpolates between a graphical Lasso and a classical Lasso, thereby establishing a connection between iOT and graph estimation, an important problem in ML.

In staggered treatment-adoption designs, later-treated units are valid controls for an earlier-treated cohort only until their own treatment begins, so the admissible donor set contracts with event time. Fixing the donor pool at the longest horizon discards temporarily eligible donors, whereas re-estimating synthetic-control weights independently at each horizon can make the counterfactual unstable as donor composition changes. We propose Risk-Set Transported Synthetic Control with Difference-in-Differences Adjustment (RT-SC-DiD). For each cohort and event-time horizon, the estimator fits weights on the currently untreated donors while shrinking them toward a transported reference that reallocates the weight of exiting donors to similar surviving donors. A DiD baseline correction removes persistent level differences. We characterize distortion from horizon-by-horizon reoptimization, derive the loading change induced by naive deletion and renormalization, and give a conditional recursive bound for error propagation under an explicitly assumed regularity condition on the transport map. We also introduce donor-support diagnostics and a donor-only placebo procedure for selecting the transport penalty. In an 80-replication pilot comparison and a separate 40-replication-per-value sensitivity analysis, intermediate transport regularization reduces average RMSE relative to independent horizon-specific estimation and strong anchoring. This evidence supports the method's bias-variance motivation but is not a proved guarantee. RT-SC-DiD is intended for settings where later-treated units provide useful short-horizon information and donor support contracts materially over time. Existing staggered synthetic-control and synthetic difference-in-differences methods do not, to our knowledge, explicitly regularize within-cohort weight sequences toward transported references as risk sets contract.

We present rectified flow, a surprisingly simple approach to learning (neural) ordinary differential equation (ODE) models to transport between two empirically observed distributions \pi_0 and \pi_1, hence providing a unified solution to generative modeling and domain transfer, among various other tasks involving distribution transport. The idea of rectified flow is to learn the ODE to follow the straight paths connecting the points drawn from \pi_0 and \pi_1 as much as possible. This is achieved by solving a straightforward nonlinear least squares optimization problem, which can be easily scaled to large models without introducing extra parameters beyond standard supervised learning. The straight paths are special and preferred because they are the shortest paths between two points, and can be simulated exactly without time discretization and hence yield computationally efficient models. We show that the procedure of learning a rectified flow from data, called rectification, turns an arbitrary coupling of \pi_0 and \pi_1 to a new deterministic coupling with provably non-increasing convex transport costs. In addition, recursively applying rectification allows us to obtain a sequence of flows with increasingly straight paths, which can be simulated accurately with coarse time discretization in the inference phase. In empirical studies, we show that rectified flow performs superbly on image generation, image-to-image translation, and domain adaptation. In particular, on image generation and translation, our method yields nearly straight flows that give high quality results even with a single Euler discretization step.

A selector that allocates work across opaque components must evaluate them without being told how they performed. We study the least communication that suffices. Each parent maintains an allocation vector over its children and updates it from outcomes by proportional redistribution. Each child reads the sign of its own allocation change, one bit per round it is selected, so no evaluation message crosses the parent-child boundary. (1) The update preserves the simplex with strict positivity. (2) Conditional on being selected, a child's sign is exactly the root outcome at every depth, a sufficient statistic. The conditioning is necessary. (3) Under interiority a single selector has a unique interior equilibrium, exact and closed-form for N=2, with almost sure convergence under decreasing steps. For general N an equi-ratio condition gives an explicit affine equilibrium. (4) The linearised mean flow has real, strictly negative tangent eigenvalues with explicit bounds for all N >= 2. (5) The mean flow converges globally. An explicit potential, up to normalisation the KL divergence between the failure and slack distributions, decreases at a rate equal to the variance of the equi-ratio statistic. Without interiority the rule drops exactly the children below the surviving support's threshold. (6) The update law composes along a single active path, since ancestors select a node's active rounds but do not alter a realised signal. No joint convergence theorem for a hierarchy whose levels adapt simultaneously is claimed. (7) The one-bit interface admits exactly four deterministic memoryless distortions, and at a single selector passing the bit on unchanged delivers strictly higher mean quality upward than negation or constant transmission. Illustrations on synthetic hierarchies up to 16,384 leaves and three real datasets (3.4 billion comparisons, no mismatches) are consistent with the theory.

We present Generative Anchored Fields (GAF), a generative model that learns independent endpoint predictors, J (noise) and K (data), from any point on a linear bridge. Unlike existing approaches that use a single trajectory or score predictor, GAF is trained to recover the bridge endpoints directly via coordinate learning. The velocity field v=K-J emerges from their time-conditioned disagreement. This factorization enables Transport Algebra: algebraic operations on multiple J/K heads for compositional control. With class-specific K_n heads, GAF defines directed transport maps between a shared base noise distribution and multiple data domains, allowing controllable interpolation, multi-class composition, and semantic editing. This is achieved either directly on the predicted data coordinates (K) using Iterative Endpoint Refinement (IER), a novel sampler that achieves high-quality generation in 5-8 steps, or on the emergent velocity field (v). We achieve strong sample quality (FID 7.51 on ImageNet 256times256 and 7.27 on CelebA-HQ 256times 256, without classifier-free guidance) while treating compositional generation as an architectural primitive. Code available at https://github.com/IDLabMedia/GAF.

With the discovery of Wasserstein GANs, Optimal Transport (OT) has become a powerful tool for large-scale generative modeling tasks. In these tasks, OT cost is typically used as the loss for training GANs. In contrast to this approach, we show that the OT map itself can be used as a generative model, providing comparable performance. Previous analogous approaches consider OT maps as generative models only in the latent spaces due to their poor performance in the original high-dimensional ambient space. In contrast, we apply OT maps directly in the ambient space, e.g., a space of high-dimensional images. First, we derive a min-max optimization algorithm to efficiently compute OT maps for the quadratic cost (Wasserstein-2 distance). Next, we extend the approach to the case when the input and output distributions are located in the spaces of different dimensions and derive error bounds for the computed OT map. We evaluate the algorithm on image generation and unpaired image restoration tasks. In particular, we consider denoising, colorization, and inpainting, where the optimality of the restoration map is a desired attribute, since the output (restored) image is expected to be close to the input (degraded) one.

Optimal Transport (OT) problem aims to find a transport plan that bridges two distributions while minimizing a given cost function. OT theory has been widely utilized in generative modeling. In the beginning, OT distance has been used as a measure for assessing the distance between data and generated distributions. Recently, OT transport map between data and prior distributions has been utilized as a generative model. These OT-based generative models share a similar adversarial training objective. In this paper, we begin by unifying these OT-based adversarial methods within a single framework. Then, we elucidate the role of each component in training dynamics through a comprehensive analysis of this unified framework. Moreover, we suggest a simple but novel method that improves the previously best-performing OT-based model. Intuitively, our approach conducts a gradual refinement of the generated distribution, progressively aligning it with the data distribution. Our approach achieves a FID score of 2.51 on CIFAR-10 and 5.99 on CelebA-HQ-256, outperforming unified OT-based adversarial approaches.

Neural posterior estimation has emerged as a powerful tool for amortized inference, with growing adoption across scientific and applied domains. In many of these applications, the conditioning variable is a set of observations whose elements depend not only on the target but also on unknown factors shared across the set. Optimal inference therefore requires treating the set jointly, which in turn requires training the estimator at the deployment set size -- a regime where memory and compute quickly become prohibitive. We introduce a simple, theoretically grounded strategy that decouples representation learning from posterior modeling. Our method trains a mean-pool Deep Set on sets of size at most two, producing an encoder that generalizes to arbitrary set sizes. The inference head is then finetuned on pre-aggregated embeddings, making training cost essentially independent of the deployment set size N. Across scalar, image, multi-view 3D, molecular, and high-dimensional conditional generation benchmarks with N in the thousands, our approach matches or outperforms standard baselines at a fraction of the compute.

Text-conditioned generative models for volumetric medical imaging provide semantic control but lack explicit anatomical guidance, often resulting in outputs that are spatially ambiguous or anatomically inconsistent. In contrast, structure-driven methods ensure strong anatomical consistency but typically assume access to ground-truth annotations, which are unavailable when the target image is to be synthesized. We propose a retrieval-augmented approach for Text-to-CT generation that integrates semantic and anatomical information under a realistic inference setting. Given a radiology report, our method retrieves a semantically related clinical case using a 3D vision-language encoder and leverages its associated anatomical annotation as a structural proxy. This proxy is injected into a text-conditioned latent diffusion model via a ControlNet branch, providing coarse anatomical guidance while maintaining semantic flexibility. Experiments on the CT-RATE dataset show that retrieval-augmented generation improves image fidelity and clinical consistency compared to text-only baselines, while additionally enabling explicit spatial controllability, a capability inherently absent in such approaches. Further analysis highlights the importance of retrieval quality, with semantically aligned proxies yielding consistent gains across all evaluation axes. This work introduces a principled and scalable mechanism to bridge semantic conditioning and anatomical plausibility in volumetric medical image synthesis. Code will be released.

Inferring cellular dynamics from unpaired single-cell snapshots requires modeling both state transitions and population growth or death. Unbalanced dynamic optimal transport (UDOT) addresses this by penalizing growth along transport paths, making the choice of growth penalty a key way to encode biological priors on proliferation and apoptosis. However, existing UDOT solvers either rely on computationally expensive NeuralODE simulations or depend on analytical solutions of conditional paths, restricting their efficiency solely to quadratic penalties, i.e. Wasserstein-Fisher-Rao (WFR) geodesics. To enable an efficient UDOT solver for general growth penalties, we first show that concave growth penalties lead to degenerate solutions where growth and transport are separated. We then introduce Simulation-free Unbalanced Dynamic Optimal transport (SUDO), a simulation-free framework for UDOT with general non-quadratic convex growth penalties. SUDO learns the conditional paths and transport costs, solves the induced semi-coupling problem, and subsequently leverages unbalanced flow matching to achieve a simulation-free solution. On WFR benchmarks, SUDO matches the accuracy of efficient, analytical solution-driven algorithms while outperforming simulation-based methods in computational speed. Beyond WFR, SUDO supports asymmetric penalties that encode proliferation-dominant priors and produce more plausible trajectories and growth estimates on synthetic and single-cell datasets.

Optimal transport is an important tool in machine learning, allowing to capture geometric properties of the data through a linear program on transport polytopes. We present a single-loop optimization algorithm for minimizing general convex objectives on these domains, utilizing the principles of Sinkhorn matrix scaling and mirror descent. The proposed algorithm is robust to noise, and can be used in an online setting. We provide theoretical guarantees for convex objectives and experimental results showcasing it effectiveness on both synthetic and real-world data.

Differentiable matching layers and residual connection paradigms, often implemented via entropy-regularized Optimal Transport (OT), serve as critical mechanisms in structural prediction and architectural scaling. However, recovering discrete permutations or maintaining identity mappings via annealing εto 0 is notoriously unstable. In this work, we identify a fundamental mechanism for this failure: Premature Mode Collapse. By analyzing the non-normal dynamics of the Sinkhorn fixed-point map, we reveal a theoretical thermodynamic speed limit: standard exponential cooling outpaces the contraction rate of the inference operator, which degrades as O(1/ε). To address this, we propose Efficient Piecewise Hybrid Adaptive Stability Control (EPH-ASC), an adaptive scheduling algorithm that monitors the stability of the inference process. We demonstrate that EPH-ASC is essential for stabilizing Manifold-Constrained Hyper-Connections (mHC) during large-scale training on the FineWeb-Edu dataset, effectively preventing late-stage gradient explosions by enforcing a linear stability law.

While conditional diffusion models are known to have good coverage of the data distribution, they still face limitations in output diversity, particularly when sampled with a high classifier-free guidance scale for optimal image quality or when trained on small datasets. We attribute this problem to the role of the conditioning signal in inference and offer an improved sampling strategy for diffusion models that can increase generation diversity, especially at high guidance scales, with minimal loss of sample quality. Our sampling strategy anneals the conditioning signal by adding scheduled, monotonically decreasing Gaussian noise to the conditioning vector during inference to balance diversity and condition alignment. Our Condition-Annealed Diffusion Sampler (CADS) can be used with any pretrained model and sampling algorithm, and we show that it boosts the diversity of diffusion models in various conditional generation tasks. Further, using an existing pretrained diffusion model, CADS achieves a new state-of-the-art FID of 1.70 and 2.31 for class-conditional ImageNet generation at 256times256 and 512times512 respectively.

Conditional diffusion and flow models routinely fail to satisfy the very constraints that define their task. For instance, a depth-conditioned model often produces images whose re-extracted depth disagrees with the input, even though the forward operator--the depth predictor defining the constraint--is available during both training and inference. Existing approaches generally fall into two categories: supervised models that treat the conditioning signal as a static cue and ignore alignment information at inference, and guidance-based methods that consult it through hand-tuned linear updates, typically trading fidelity to the condition against the plausibility of the generated sample. We argue that the fundamental gap in both paradigms is that the model is never trained to utilize its own alignment error. We introduce FlowBender, a closed-loop framework that treats this error as a first-class input, training the network to learn a correction policy conditioned on inference-time feedback. At each step, an unguided look-ahead pass estimates the clean signal, a task-specific deviation is computed via the forward operator, and a refinement pass consumes this signal to produce a corrected velocity. We propose several variants of FlowBender, including a gradient-based formulation for differentiable operators and a zero-order variant for non-differentiable settings such as JPEG compression. For efficient sampling, we introduce a prior-step shortcut that enables closed-loop correction at a minimal additional computational cost. Across image-to-image translation, restoration, and 3D mesh texturing, FlowBender consistently outperforms standard supervised baselines, alignment-loss-augmented training, and state-of-the-art inference-time guidance, improving fidelity and plausibility simultaneously rather than trading them against each other. Project page: https://flow-bender.github.io/

The recent emergence of new algorithms for permuting models into functionally equivalent regions of the solution space has shed some light on the complexity of error surfaces, and some promising properties like mode connectivity. However, finding the right permutation is challenging, and current optimization techniques are not differentiable, which makes it difficult to integrate into a gradient-based optimization, and often leads to sub-optimal solutions. In this paper, we propose a Sinkhorn re-basin network with the ability to obtain the transportation plan that better suits a given objective. Unlike the current state-of-art, our method is differentiable and, therefore, easy to adapt to any task within the deep learning domain. Furthermore, we show the advantage of our re-basin method by proposing a new cost function that allows performing incremental learning by exploiting the linear mode connectivity property. The benefit of our method is compared against similar approaches from the literature, under several conditions for both optimal transport finding and linear mode connectivity. The effectiveness of our continual learning method based on re-basin is also shown for several common benchmark datasets, providing experimental results that are competitive with state-of-art results from the literature.

Transductive Few-Shot Learning (TFSL) has recently attracted increasing attention since it typically outperforms its inductive peer by leveraging statistics of query samples. However, previous TFSL methods usually encode uniform prior that all the classes within query samples are equally likely, which is biased in imbalanced TFSL and causes severe performance degradation. Given this pivotal issue, in this work, we propose a novel Conditional Transport (CT) based imbalanced TFSL model called {\textbf P}rototypes-oriented {\textbf U}nbiased {\textbf T}ransfer {\textbf M}odel (PUTM) to fully exploit unbiased statistics of imbalanced query samples, which employs forward and backward navigators as transport matrices to balance the prior of query samples per class between uniform and adaptive data-driven distributions. For efficiently transferring statistics learned by CT, we further derive a closed form solution to refine prototypes based on MAP given the learned navigators. The above two steps of discovering and transferring unbiased statistics follow an iterative manner, formulating our EM-based solver. Experimental results on four standard benchmarks including miniImageNet, tieredImageNet, CUB, and CIFAR-FS demonstrate superiority of our model in class-imbalanced generalization.

We propose a unified, few-step generative modeling framework based on cumulative flow maps for long-range transport in probability space, inspired by flow-map techniques for physical transport and dynamics. At its core is a cumulative-flow abstraction that connects local, instantaneous updates with finite-time transport, enabling generative models to reason about global state transitions. This perspective yields a unified few-step framework built on cumulative transport and cumulative parameterization that applies broadly to existing diffusion- and flow-based models without being tied to a specific prediction instantiation. Our formulation supports few-step and even one-step generation while preserving synthesis quality, requiring only minimal changes to time embeddings and training objectives, and no increase in model capacity. We demonstrate its effectiveness across diverse tasks, including image generation, geometric distribution modeling, joint prediction, and SDF generation, with reduced inference cost.

Fluorescence microscopy reveals where proteins localize, but only a limited number of proteins can be imaged in the same cell; generating these images from amino-acid sequence and the cell's morphological context enables in silico localization of unimaged proteins. The two conditions, however, play asymmetric roles: morphological context is spatially aligned with the target, whereas sequence is non-spatial and must specify protein-dependent localization within it, with recurring coarse patterns shared across proteins and finer protein-specific variation. Existing generators condition on both jointly, without separating what each explains. We introduce FACET (Factorized Asymmetric Conditioning for Efficient Transport), a probabilistic generative framework that encodes this structure as an explicit inductive bias: sequence semantics are learned from what context leaves unexplained, coarse localization regularities are shared across proteins through a semantic memory, and protein-specific variation is a bounded residual around them. A variance-preserving state projection further lets FACET perform continuous stochastic transport through a pretrained diffusion predictor with minimal parameter overhead. On held-out proteins, FACET improves spatial overlap by 34.3% on the Human Protein Atlas and 14.0% on OpenCell over a backbone-matched baseline, and reduces FID by 27.2% and 46.5%, respectively, with 75% fewer network evaluations. It also substantially improves protein-association structure recovery and yields better-calibrated predictions, while detailed ablations show complementary contributions from its design choices. These results identify factorized asymmetric conditioning, rather than generator capacity alone, as a key lever for high-fidelity, efficient, and biologically meaningful cellular image synthesis.

Guidance of generative models is typically achieved by modifying the probability flow vector field through the addition of a guidance field. In this paper, we instead propose the Source-Guided Flow Matching (SGFM) framework, which modifies the source distribution directly while keeping the pre-trained vector field intact. This reduces the guidance problem to a well-defined problem of sampling from the source distribution. We theoretically show that SGFM recovers the desired target distribution exactly. Furthermore, we provide bounds on the Wasserstein error for the generated distribution when using an approximate sampler of the source distribution and an approximate vector field. The key benefit of our approach is that it allows the user to flexibly choose the sampling method depending on their specific problem. To illustrate this, we systematically compare different sampling methods and discuss conditions for asymptotically exact guidance. Moreover, our framework integrates well with optimal flow matching models since the straight transport map generated by the vector field is preserved. Experimental results on synthetic 2D benchmarks, physics-informed generative tasks, and imaging inverse problems demonstrate the effectiveness and flexibility of the proposed framework.

In many real-world scenarios, obtaining fully observed samples is prohibitively expensive or even infeasible, while partial and noisy observations are comparatively easy to collect. In this work, we study distribution restoration with abundant noisy samples, assuming the corruption process is available as a black-box generator. We show that this task can be framed as a one-sided entropic optimal transport problem and solved via an EM-like algorithm. We further provide a test criterion to determine whether the true underlying distribution is recoverable under per-sample information loss, and show that in otherwise unrecoverable cases, a small number of clean samples can render the distribution largely recoverable. Building on these insights, we introduce SFBD-OMNI, a bridge model-based framework that maps corrupted sample distributions to the ground-truth distribution. Our method generalizes Stochastic Forward-Backward Deconvolution (SFBD; Lu et al., 2025) to handle arbitrary measurement models beyond Gaussian corruption. Experiments across benchmark datasets and diverse measurement settings demonstrate significant improvements in both qualitative and quantitative performance.

Deep learning models have emerged as a powerful tool for various medical applications. However, their success depends on large, high-quality datasets that are challenging to obtain due to privacy concerns and costly annotation. Generative models, such as diffusion models, offer a potential solution by synthesizing medical images, but their practical adoption is hindered by long inference times. In this paper, we propose the use of an optimal transport flow matching approach to accelerate image generation. By introducing a straighter mapping between the source and target distribution, our method significantly reduces inference time while preserving and further enhancing the quality of the outputs. Furthermore, this approach is highly adaptable, supporting various medical imaging modalities, conditioning mechanisms (such as class labels and masks), and different spatial dimensions, including 2D and 3D. Beyond image generation, it can also be applied to related tasks such as image enhancement. Our results demonstrate the efficiency and versatility of this framework, making it a promising advancement for medical imaging applications. Code with checkpoints and a synthetic dataset (beneficial for classification and segmentation) is now available on: https://github.com/milad1378yz/MOTFM.

Many scientific questions require reasoning about what was never observed: What if the conditions, interventions, or history had been different? Models can predict accurately on observed data yet fail on such what-if queries when correlated inputs are varied independently. A common remedy is to add controlled simulation data in which these factors are explicitly disentangled, but this requires access to a simulator, can be computationally expensive, and inherits the simulator's modeling assumptions. We introduce ReRoute, a framework for targeted scientific what-if prediction that combines factual data with partial mechanistic knowledge, without requiring controlled intervention data for adaptation. ReRoute fixes the queried input of a pretrained backbone to a reference value, reintroduces its variation through a known mechanistic pathway, and fine-tunes on the original factual data, while leaving downstream effects to the learned dynamics. We provide a causal identification result for this construction under explicit structural assumptions, with the core argument machine-checked in Lean. After showing that ReRoute achieves highly accurate counterfactual predictions in a controlled advection-diffusion system where exact responses are available, we turn to state-of-the-art climate emulation. On held-out coupled-climate interventions, ReRoute reduces aggregate climate error by 18.2-31.8% under severe CO_2 distribution shifts while preserving skill under standard conditions, at a small fraction of the cost of retraining on additional controlled simulations, without even accounting for the substantial expense of generating such data. Finally, on an emulator trained from historical ERA5 reanalysis, where no counterfactual reference exists, ReRoute preserves substantially more of the surface warming implied by the observed boundary conditions under a fixed-CO_2 counterfactual.

We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. This helps repeatedly solve similar OT problems between different measures by leveraging the knowledge and information present from past problems to rapidly predict and solve new problems. Otherwise, standard methods ignore the knowledge of the past solutions and suboptimally re-solve each problem from scratch. We instantiate Meta OT models in discrete and continuous settings between grayscale images, spherical data, classification labels, and color palettes and use them to improve the computational time of standard OT solvers. Our source code is available at http://github.com/facebookresearch/meta-ot

Optimal transport (OT) has gained popularity due to its various applications in fields such as machine learning, statistics, and signal processing. However, the balanced mass requirement limits its performance in practical problems. To address these limitations, variants of the OT problem, including unbalanced OT, Optimal partial transport (OPT), and Hellinger Kantorovich (HK), have been proposed. In this paper, we propose the Linear optimal partial transport (LOPT) embedding, which extends the (local) linearization technique on OT and HK to the OPT problem. The proposed embedding allows for faster computation of OPT distance between pairs of positive measures. Besides our theoretical contributions, we demonstrate the LOPT embedding technique in point-cloud interpolation and PCA analysis.

Generative policies based on diffusion and flow matching have become a dominant paradigm for visuomotor robotic control. We show that replacing the standard Gaussian source distribution with WarmPrior, a simple temporally grounded prior constructed from readily available recent action history, consistently improves success rates on robotic manipulation tasks. We trace this gain to markedly straighter probability paths, echoing the effect of optimal-transport couplings in Rectified Flow. Beyond standard behavior cloning, WarmPrior also reshapes the exploration distribution in prior-space reinforcement learning, improving both sample efficiency and final performance. Collectively, these results identify the source distribution as an important and underexplored design axis in generative robot control.

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.

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another. Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the ``best'' map to morph a continuous measure in P(Rd) into another must be the gradient of a convex function. To exploit that result, [Makkuva+ 2020, Korotin+2020] consider maps T=nabla f_theta, where f_theta is an input convex neural network (ICNN), as defined by Amos+2017, and fit theta with SGD using samples. Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on theta; the need to approximate the conjugate of f_theta; or the limitation that they only work for the squared-Euclidean cost. More generally, we question the relevance of using Brenier's result, which only applies to densities, to constrain the architecture of candidate maps fitted on samples. Motivated by these limitations, we propose a radically different approach to estimating OT maps: Given a cost c and a reference measure rho, we introduce a regularizer, the Monge gap M^c_{rho}(T) of a map T. That gap quantifies how far a map T deviates from the ideal properties we expect from a c-OT map. In practice, we drop all architecture requirements for T and simply minimize a distance (e.g., the Sinkhorn divergence) between Tsharpmu and nu, regularized by M^c_rho(T). We study M^c_{rho}, and show how our simple pipeline outperforms significantly other baselines in practice.

Traditional learning systems are trained in closed-world for a fixed number of classes, and need pre-collected datasets in advance. However, new classes often emerge in real-world applications and should be learned incrementally. For example, in electronic commerce, new types of products appear daily, and in a social media community, new topics emerge frequently. Under such circumstances, incremental models should learn several new classes at a time without forgetting. We find a strong correlation between old and new classes in incremental learning, which can be applied to relate and facilitate different learning stages mutually. As a result, we propose CO-transport for class Incremental Learning (COIL), which learns to relate across incremental tasks with the class-wise semantic relationship. In detail, co-transport has two aspects: prospective transport tries to augment the old classifier with optimal transported knowledge as fast model adaptation. Retrospective transport aims to transport new class classifiers backward as old ones to overcome forgetting. With these transports, COIL efficiently adapts to new tasks, and stably resists forgetting. Experiments on benchmark and real-world multimedia datasets validate the effectiveness of our proposed method.

Vector-quantized networks (VQNs) have exhibited remarkable performance across various tasks, yet they are prone to training instability, which complicates the training process due to the necessity for techniques such as subtle initialization and model distillation. In this study, we identify the local minima issue as the primary cause of this instability. To address this, we integrate an optimal transport method in place of the nearest neighbor search to achieve a more globally informed assignment. We introduce OptVQ, a novel vector quantization method that employs the Sinkhorn algorithm to optimize the optimal transport problem, thereby enhancing the stability and efficiency of the training process. To mitigate the influence of diverse data distributions on the Sinkhorn algorithm, we implement a straightforward yet effective normalization strategy. Our comprehensive experiments on image reconstruction tasks demonstrate that OptVQ achieves 100% codebook utilization and surpasses current state-of-the-art VQNs in reconstruction quality.

Unconditional flow-matching trains diffusion models to transport samples from a source distribution to a target distribution by enforcing that the flows between sample pairs are unique. However, in conditional settings (e.g., class-conditioned models), this uniqueness is no longer guaranteed--flows from different conditions may overlap, leading to more ambiguous generations. We introduce Contrastive Flow Matching, an extension to the flow matching objective that explicitly enforces uniqueness across all conditional flows, enhancing condition separation. Our approach adds a contrastive objective that maximizes dissimilarities between predicted flows from arbitrary sample pairs. We validate Contrastive Flow Matching by conducting extensive experiments across varying model architectures on both class-conditioned (ImageNet-1k) and text-to-image (CC3M) benchmarks. Notably, we find that training models with Contrastive Flow Matching (1) improves training speed by a factor of up to 9x, (2) requires up to 5x fewer de-noising steps and (3) lowers FID by up to 8.9 compared to training the same models with flow matching. We release our code at: https://github.com/gstoica27/DeltaFM.git.

We introduce a novel neural network-based algorithm to compute optimal transport (OT) plans for general cost functionals. In contrast to common Euclidean costs, i.e., ell^1 or ell^2, such functionals provide more flexibility and allow using auxiliary information, such as class labels, to construct the required transport map. Existing methods for general costs are discrete and have limitations in practice, i.e. they do not provide an out-of-sample estimation. We address the challenge of designing a continuous OT approach for general costs that generalizes to new data points in high-dimensional spaces, such as images. Additionally, we provide the theoretical error analysis for our recovered transport plans. As an application, we construct a cost functional to map data distributions while preserving the class-wise structure.

Common image-to-image translation methods rely on joint training over data from both source and target domains. The training process requires concurrent access to both datasets, which hinders data separation and privacy protection; and existing models cannot be easily adapted for translation of new domain pairs. We present Dual Diffusion Implicit Bridges (DDIBs), an image translation method based on diffusion models, that circumvents training on domain pairs. Image translation with DDIBs relies on two diffusion models trained independently on each domain, and is a two-step process: DDIBs first obtain latent encodings for source images with the source diffusion model, and then decode such encodings using the target model to construct target images. Both steps are defined via ordinary differential equations (ODEs), thus the process is cycle consistent only up to discretization errors of the ODE solvers. Theoretically, we interpret DDIBs as concatenation of source to latent, and latent to target Schrodinger Bridges, a form of entropy-regularized optimal transport, to explain the efficacy of the method. Experimentally, we apply DDIBs on synthetic and high-resolution image datasets, to demonstrate their utility in a wide variety of translation tasks and their inherent optimal transport properties.

Recently, diffusion model shines as a promising backbone for the sequence modeling paradigm in offline reinforcement learning(RL). However, these works mostly lack the generalization ability across tasks with reward or dynamics change. To tackle this challenge, in this paper we propose a task-oriented conditioned diffusion planner for offline meta-RL(MetaDiffuser), which considers the generalization problem as conditional trajectory generation task with contextual representation. The key is to learn a context conditioned diffusion model which can generate task-oriented trajectories for planning across diverse tasks. To enhance the dynamics consistency of the generated trajectories while encouraging trajectories to achieve high returns, we further design a dual-guided module in the sampling process of the diffusion model. The proposed framework enjoys the robustness to the quality of collected warm-start data from the testing task and the flexibility to incorporate with different task representation method. The experiment results on MuJoCo benchmarks show that MetaDiffuser outperforms other strong offline meta-RL baselines, demonstrating the outstanding conditional generation ability of diffusion architecture.

The recent wave of one-step generative models, which compress the multi-step trajectory of diffusion via either distillation or learned flow maps, has reached an inflection point where they can generate high-quality images. Here, we ask a natural question that follows from these advances: what happens to the denoising trajectory of multi-step diffusion when generation is compressed into a single forward pass? We offer an empirical observation we call depth as time: the denoising computation that multi-step diffusion performs across sampling steps appears to unfold across the depth of a single forward pass, and can be recovered by decoding intermediate layers with the model's own output head. Most interestingly, we show that this depthwise computation depends on the transport task a flow map is trained to solve. The most surprising case is MeanFlow, where probing shorter transport intervals reveals both denoising and renoising within a single network evaluation. In contrast, generators trained without a time-indexed transport task, such as drifting models, do not exhibit the same depthwise denoising. Consequently, we show that models that exhibit the depthwise denoising phenomenon are more compressible across the layerwise computation: a MeanFlow SiT-L/2 model can be compressed by 16.6times in parameters into a single time-conditioned block. We offer an explanation for this denoise-then-renoise behavior and show that, when we treat the layerwise computation explicitly as a flow, a single time-conditioned block can be trained to denoise across layers, compressing a MeanFlow SiT-L/2 model by 16.6times in parameters. Together, these results suggest that the temporal computation of diffusion is not eliminated by one-step generation, but reorganized across network depth.

Given a set of K probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

Diffusion models have exhibited exciting capabilities in generating images and are also very promising for video creation. However, the inference speed of diffusion models is limited by the slow sampling process, restricting its use cases. The sequential denoising steps required for generating a single sample could take tens or hundreds of iterations and thus have become a significant bottleneck. This limitation is more salient for applications that are interactive in nature or require small latency. To address this challenge, we propose Partially Conditioned Patch Parallelism (PCPP) to accelerate the inference of high-resolution diffusion models. Using the fact that the difference between the images in adjacent diffusion steps is nearly zero, Patch Parallelism (PP) leverages multiple GPUs communicating asynchronously to compute patches of an image in multiple computing devices based on the entire image (all patches) in the previous diffusion step. PCPP develops PP to reduce computation in inference by conditioning only on parts of the neighboring patches in each diffusion step, which also decreases communication among computing devices. As a result, PCPP decreases the communication cost by around 70% compared to DistriFusion (the state of the art implementation of PP) and achieves 2.36sim 8.02times inference speed-up using 4sim 8 GPUs compared to 2.32sim 6.71times achieved by DistriFusion depending on the computing device configuration and resolution of generation at the cost of a possible decrease in image quality. PCPP demonstrates the potential to strike a favorable trade-off, enabling high-quality image generation with substantially reduced latency.

A common recipe to improve diffusion models at test-time so that samples score highly against a user-specified reward is to introduce the gradient of the reward into the dynamics of the diffusion itself. This procedure is often ill posed, as user-specified rewards are usually only well defined on the data distribution at the end of generation. While common workarounds to this problem are to use a denoiser to estimate what a sample would have been at the end of generation, we propose a simple solution to this problem by working directly with a flow map. By exploiting a relationship between the flow map and velocity field governing the instantaneous transport, we construct an algorithm, Flow Map Trajectory Tilting (FMTT), which provably performs better ascent on the reward than standard test-time methods involving the gradient of the reward. The approach can be used to either perform exact sampling via importance weighting or principled search that identifies local maximizers of the reward-tilted distribution. We demonstrate the efficacy of our approach against other look-ahead techniques, and show how the flow map enables engagement with complicated reward functions that make possible new forms of image editing, e.g. by interfacing with vision language models.

Probabilistic weather forecasting requires not only accurate trajectories, but calibrated distributions over plausible atmospheric futures. Recent data-driven systems have achieved remarkable deterministic skill, and diffusion-based ensemble forecasters have substantially improved sample realism and uncertainty quantification. However, their inference cost scales with forecast horizon, ensemble size, and the number of denoising steps required for each transition, making large operational ensembles expensive. To address this, we present Tyche, a one-step conditional flow model for efficient probabilistic weather forecasting. Tyche models the conditional forecast distribution with a destination-aware average-velocity flow that maps Gaussian noise directly to future weather states in a single function evaluation (1-NFE). To make this one-step transport learnable in high-dimensional geophysical fields, we derive a JVP-regularized rectification objective that enforces temporal self-consistency across source and destination flow timesteps without explicitly forming Jacobians. The transport field is parameterized by an isotropic Swin-style transformer that preserves fine-scale spatial structure while remaining scalable on global grids. To improve ensemble reliability under autoregressive forecasting, we further introduce a rollout-based finetuning stage with curriculum CRPS calibration supervision. Experiments on ERA5 at 1.5^circ and 6-hour resolution show that our Tyche, using merely a single NFE, matches or exceeds the forecast skill and calibration of state-of-the-art multi-step generative baselines and the operational ECMWF IFS ensemble.

Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

Traditionally, data selection has been studied in settings where all samples from prospective sources are fully revealed to a machine learning developer. However, in practical data exchange scenarios, data providers often reveal only a limited subset of samples before an acquisition decision is made. Recently, there have been efforts to fit scaling laws that predict model performance at any size and data source composition using the limited available samples. However, these scaling functions are black-box, computationally expensive to fit, highly susceptible to overfitting, or/and difficult to optimize for data selection. This paper proposes a framework called <projektor>, which predicts model performance and supports data selection decisions based on partial samples of prospective data sources. Our approach distinguishes itself from existing work by introducing a novel *two-stage* performance inference process. In the first stage, we leverage the Optimal Transport distance to predict the model's performance for any data mixture ratio within the range of disclosed data sizes. In the second stage, we extrapolate the performance to larger undisclosed data sizes based on a novel parameter-free mapping technique inspired by neural scaling laws. We further derive an efficient gradient-based method to select data sources based on the projected model performance. Evaluation over a diverse range of applications demonstrates that <projektor> significantly improves existing performance scaling approaches in terms of both the accuracy of performance inference and the computation costs associated with constructing the performance predictor. Also, <projektor> outperforms by a wide margin in data selection effectiveness compared to a range of other off-the-shelf solutions.

A distribution shift can have fundamental consequences such as signaling a change in the operating environment or significantly reducing the accuracy of downstream models. Thus, understanding distribution shifts is critical for examining and hopefully mitigating the effect of such a shift. Most prior work focuses on merely detecting if a shift has occurred and assumes any detected shift can be understood and handled appropriately by a human operator. We hope to aid in these manual mitigation tasks by explaining the distribution shift using interpretable transportation maps from the original distribution to the shifted one. We derive our interpretable mappings from a relaxation of optimal transport, where the candidate mappings are restricted to a set of interpretable mappings. We then inspect multiple quintessential use-cases of distribution shift in real-world tabular, text, and image datasets to showcase how our explanatory mappings provide a better balance between detail and interpretability than baseline explanations by both visual inspection and our PercentExplained metric.

We develop and analyse a conforming piecewise affine finite element method for the Grad--Mercier equation, a nonlocal elliptic model for magnetic equilibria whose source depends on the global rank of a flux function. The principal difficulty is that the unique continuum equilibrium has a maximum plateau of prescribed positive area. The solution therefore samples the rank law only above the plateau mass, while perturbations that separate equal values may distribute this mass over several numerical ranks and thereby alter the non-strict rank source. We exploit the continuum well-posedness structure to replace the rank law by an equivalent clipped, nonnegative law, and replace Lebesgue measure by a mass-lumped nodal measure. Equal nodal values are assigned a common rank interval, thereby retaining the full lumped mass of each tied level. The resulting weighted rearrangement gives a strictly convex discrete energy and hence a unique finite element equilibrium. On meshes satisfying a discrete comparison principle, the discrete equilibrium is nonnegative and bounded above by a computable barrier. A Poisson-preconditioned rearrangement iteration converges on each fixed mesh and admits computable a posteriori energy and energy-norm bounds. As polygonal finite element domains converge to the physical cross-section, the discrete equilibria converge strongly in H^1_0(Ω) and uniformly on overlineΩ to the unique Grad--Mercier equilibrium. Reconstructed equilibrium sources converge weak-* in L^{infty}(Ω) to the original nonlocal source.

Effective image inversion in rectified flow models - mapping real images to editable latent representations - is crucial for practical image editing applications; however, achieving optimal balance between reconstruction fidelity and editing flexibility remains a fundamental challenge. In this work, we introduce the Optimal Transport Inversion Pipeline (OTIP), a zero-shot framework that leverages optimal transport theory to guide the inversion process in rectified flow models. Our underlying hypothesis is that incorporating transport-based guidance during the reverse diffusion process can effectively balance reconstruction accuracy and editing controllability through principled trajectory optimization. The method computes optimal transport paths between image and noise distributions while maintaining computational efficiency. Our approach achieves high-fidelity reconstruction with LPIPS scores of 0.001 and SSIM of 0.992 on face editing benchmarks, demonstrating superior preservation of fine-grained details compared to existing methods. We evaluate the framework across multiple editing tasks, observing 7.8% to 12.9% improvements in reconstruction loss over RF-Inversion on the LSUN-Bedroom and LSUN-Church datasets, respectively. For semantic face editing, our method achieves an 11.2% improvement in identity preservation and a 1.6% enhancement in perceptual quality, while maintaining computational efficiency comparable to baseline approaches. Qualitatively, our method produces visually compelling edits with superior semantic consistency and fine-grained detail preservation across diverse editing scenarios. Code is available at: https://github.com/marianlupascu/OT-Inversion

Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with a very high number of supports. The m-OT solves several smaller optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, the m-OT does not consider the relationship between mini-batches which leads to undesirable estimation. Moreover, the m-OT does not approximate a proper metric between probability measures since the identity property is not satisfied. To address these problems, we propose a novel mini-batch scheme for optimal transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that finds the optimal coupling between mini-batches and it can be seen as an approximation to a well-defined distance on the space of probability measures. Furthermore, we show that the m-OT is a limit of the entropic regularized version of the BoMb-OT when the regularized parameter goes to infinity. Finally, we carry out experiments on various applications including deep generative models, deep domain adaptation, approximate Bayesian computation, color transfer, and gradient flow to show that the BoMb-OT can be widely applied and performs well in various applications.

A common architectural choice for deep metric learning is a convolutional neural network followed by global average pooling (GAP). Albeit simple, GAP is a highly effective way to aggregate information. One possible explanation for the effectiveness of GAP is considering each feature vector as representing a different semantic entity and GAP as a convex combination of them. Following this perspective, we generalize GAP and propose a learnable generalized sum pooling method (GSP). GSP improves GAP with two distinct abilities: i) the ability to choose a subset of semantic entities, effectively learning to ignore nuisance information, and ii) learning the weights corresponding to the importance of each entity. Formally, we propose an entropy-smoothed optimal transport problem and show that it is a strict generalization of GAP, i.e., a specific realization of the problem gives back GAP. We show that this optimization problem enjoys analytical gradients enabling us to use it as a direct learnable replacement for GAP. We further propose a zero-shot loss to ease the learning of GSP. We show the effectiveness of our method with extensive evaluations on 4 popular metric learning benchmarks. Code is available at: GSP-DML Framework

Editing real images using a pre-trained text-to-image (T2I) diffusion/flow model often involves inverting the image into its corresponding noise map. However, inversion by itself is typically insufficient for obtaining satisfactory results, and therefore many methods additionally intervene in the sampling process. Such methods achieve improved results but are not seamlessly transferable between model architectures. Here, we introduce FlowEdit, a text-based editing method for pre-trained T2I flow models, which is inversion-free, optimization-free and model agnostic. Our method constructs an ODE that directly maps between the source and target distributions (corresponding to the source and target text prompts) and achieves a lower transport cost than the inversion approach. This leads to state-of-the-art results, as we illustrate with Stable Diffusion 3 and FLUX. Code and examples are available on the project's webpage.

Video creation spans text-to-video (T2V), image-to-video (I2V), and condition-based generation, yet video diffusion models remain costly because they repeatedly evaluate large backbones during sampling. Distribution matching distillation (DMD) reduces this cost, but its reverse Kullback--Leibler (KL) objective can provide unstable or incomplete guidance when the student and teacher distributions have limited overlap. VDOT addressed this issue by adding optimal transport distillation (OTD), whose explicit coupling supplies geometric directions for condition-based generation. Balanced OTD, however, performs full-mass matching between the spatial tokens of each corresponding student--teacher frame pair. This assumption weakens for T2V and I2V, where one condition admits many valid outputs and spatial content need not align across different realizations. We present VDOT++, a unified distillation framework that applies the same training recipe separately to generators for the three task families. It makes OTD robust to output diversity through an asymmetric unbalanced formulation that allows unreliable student tokens to carry less mass while maintaining coverage of the teacher tokens. An ell_1 ground cost further replaces mean-based aggregation with a more mode-preserving weighted median that limits the influence of distant transport targets. The two changes respectively determine whom to match and how the selected targets should be aggregated. We additionally combine distribution matching and adversarial refinement through sequential backward passes, and exploit the decoupled score networks for cross-scale distillation, where larger score networks improve a compact generator. Experiments on UVCBench, VBench, VBench-I2V, and the VACE benchmark show that the resulting four-step generators are competitive with many-step teachers and strong few-step baselines across all three task families.

Diffusion models have become the go-to method for large-scale generative models in real-world applications. These applications often involve data distributions confined within bounded domains, typically requiring ad-hoc thresholding techniques for boundary enforcement. Reflected diffusion models (Lou23) aim to enhance generalizability by generating the data distribution through a backward process governed by reflected Brownian motion. However, reflected diffusion models may not easily adapt to diverse domains without the derivation of proper diffeomorphic mappings and do not guarantee optimal transport properties. To overcome these limitations, we introduce the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive elegant reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and explore natural connections to entropic optimal transport for the study of approximate linear convergence - a valuable insight for practical training. Our algorithm yields robust generative modeling in diverse domains, and its scalability is demonstrated in real-world constrained generative modeling through standard image benchmarks.

This paper introduces Gaussian Spatial Transport (GST), a novel framework that leverages Gaussian splatting to facilitate transport from the probability measure in the image coordinate space to the annotation map. We propose a Gaussian splatting-based method to estimate pixel-annotation correspondence, which is then used to compute a transport plan derived from Bayesian probability. To integrate the resulting transport plan into standard network optimization in typical computer vision tasks, we derive a loss function that measures discrepancy after transport. Extensive experiments on representative computer vision tasks, including crowd counting and landmark detection, validate the effectiveness of our approach. Compared to conventional optimal transport schemes, GST eliminates iterative transport plan computation during training, significantly improving efficiency. Code is available at https://github.com/infinite0522/GST.

A framework is proposed that addresses both conditional density estimation and latent variable discovery. The objective function maximizes explanation of variability in the data, achieved through the optimal transport barycenter generalized to a collection of conditional distributions indexed by a covariate --either given or latent-- in any suitable space. Theoretical results establish the existence of barycenters, a minimax formulation of optimal transport maps, and a general characterization of variability via the optimal transport cost. This framework leads to a family of non-parametric neural network-based algorithms, the BaryNet, with a supervised version that estimates conditional distributions and an unsupervised version that assigns latent variables. The efficacy of BaryNets is demonstrated by tests on both artificial and real-world data sets. A parallel drawn between autoencoders and the barycenter framework leads to the Barycentric autoencoder algorithm (BAE).

Modern generative models typically rely on an adversarial critic, a prescribed noise-to-data path, or an autoregressive factorization. Instead, we show that a proper distributional energy can induce sample-level motion and provide direct regression supervision for a one-step generator. Three-Body Scattering Modeling (TBSM) for generation turns the energy distance into a constant-size per-projectile interaction: each projectile is attracted toward one real source and repelled from one independently generated source. Conditioned on the projectile and its condition, its expectation equals the 2-Wasserstein gradient-flow velocity of frac12D_E^2(P_θ,Q). A batch of B frozen-target events yields O(B) sample-level losses, each using one reference for its condition instead of the minibatch-wide all-pairs field used by methods such as Drifting Models. Tracking this conditional expectation online can reduce field noise. Using scattering in frozen image features, TBSM trains one-step generators on ImageNet-256, achieving FID{}=2.23 with pixel-space PixelDiT-XL and FID{}=1.63 with latent-space DiT-XL at NFE{}=1. We provide a design map relating diffusion-related supervision, Drift-like dynamics, and GAN-like objectives. These results establish tracked scattering as a route to high-dimensional one-step generation. Code: https://github.com/sp12138/TBSM.

Diffusion models and flow-based methods have shown impressive generative capability, especially for images, but their sampling is expensive because it requires many iterative updates. We introduce W-Flow, a framework for training a generator that transforms samples from a simple reference distribution into samples from a target data distribution in a single step. This is achieved in two steps: we first define an evolution from the reference distribution to the target distribution through a Wasserstein gradient flow that minimizes an energy functional; second, we train a static neural generator to compress this evolution into one-step generation. We instantiate the energy functional with the Sinkhorn divergence, which yields an efficient optimal-transport-based update rule that captures global distributional discrepancy and improves coverage of the target distribution. We further prove that the finite-sample training dynamics converge to the continuous-time distributional dynamics under suitable assumptions. Empirically, W-Flow sets a new state of the art for one-step ImageNet 256times256 generation, achieving 1.29 FID, with improved mode coverage and domain transfer. Compared to multi-step diffusion models with similar FID scores, our method yields approximately 100times faster sampling. These results show that Wasserstein gradient flows provide a principled and effective foundation for fast and high-fidelity generative modeling.

The substantial training cost of diffusion models hinders their deployment. Immiscible Diffusion recently showed that reducing diffusion trajectory mixing in the noise space via linear assignment accelerates training by simplifying denoising. To extend immiscible diffusion beyond the inefficient linear assignment under high batch sizes and high dimensions, we refine this concept to a broader miscibility reduction at any layer and by any implementation. Specifically, we empirically demonstrate the bijective nature of the denoising process with respect to immiscible diffusion, ensuring its preservation of generative diversity. Moreover, we provide thorough analysis and show step-by-step how immiscibility eases denoising and improves efficiency. Extending beyond linear assignment, we propose a family of implementations including K-nearest neighbor (KNN) noise selection and image scaling to reduce miscibility, achieving up to >4x faster training across diverse models and tasks including unconditional/conditional generation, image editing, and robotics planning. Furthermore, our analysis of immiscibility offers a novel perspective on how optimal transport (OT) enhances diffusion training. By identifying trajectory miscibility as a fundamental bottleneck, we believe this work establishes a potentially new direction for future research into high-efficiency diffusion training. The code is available at https://github.com/yhli123/Immiscible-Diffusion.

Complete orthogonal predictive coordinates do not by themselves bind a latent direction to a named intervention. We present a mathematical and synthetic audit motivated by vascular device-vessel suitcordance. Capacity-matched least-squares predictors were exactly equivalent under complete fixed output transforms, whereas an anchor-only observer recovered interpretations only within the span of known perturbation signatures. Six three-dimensional configurations across 64 seeds gave a maximum paired prediction discrepancy of 6.7e-15 but a median untransported edit error of 1.513. Coordinate transport removed that error. Noisy and weak anchors constrained calibration stability, and changing the representation basis required recalibration or verified transport. Across 256 additional fits in dimensions 3-24, prediction equivalence persisted within 4.0e-15. We then evaluated nine deliberate runnable fault classes across 64 seeds. All 576 faulty executions completed, but each violated at least one reconstruction, prediction, delivered-edit or scope contract; all 320 valid control records passed. Repeating a faulty implementation gave exact self-agreement despite error against the separately computed simulator expectation. For one omitted-direction defect, probe coverage followed its analytic law, and rank-aware abstention protected unsupported interpretations. Scalar-noise experiments exposed both missed weak faults and excessive rejection under narrow relative tolerances. These controls provide an executable separation of prediction, semantic support and scientific acceptance. They are synthetic numerical audits, not clinical validation, neural JEPA-Anything replication, agent learning or patient treatment-effect estimation.

Computing the optimal transport distance between statistical distributions is a fundamental task in machine learning. One remarkable recent advancement is entropic regularization and the Sinkhorn algorithm, which utilizes only matrix scaling and guarantees an approximated solution with near-linear runtime. Despite the success of the Sinkhorn algorithm, its runtime may still be slow due to the potentially large number of iterations needed for convergence. To achieve possibly super-exponential convergence, we present Sinkhorn-Newton-Sparse (SNS), an extension to the Sinkhorn algorithm, by introducing early stopping for the matrix scaling steps and a second stage featuring a Newton-type subroutine. Adopting the variational viewpoint that the Sinkhorn algorithm maximizes a concave Lyapunov potential, we offer the insight that the Hessian matrix of the potential function is approximately sparse. Sparsification of the Hessian results in a fast O(n^2) per-iteration complexity, the same as the Sinkhorn algorithm. In terms of total iteration count, we observe that the SNS algorithm converges orders of magnitude faster across a wide range of practical cases, including optimal transportation between empirical distributions and calculating the Wasserstein W_1, W_2 distance of discretized densities. The empirical performance is corroborated by a rigorous bound on the approximate sparsity of the Hessian matrix.

Three-dimensional (3D) molecule generation has been dominated by diffusion models, which achieve strong generation quality but typically require the molecular size to be specified a priori. Recent autoregressive approaches have substantially narrowed the performance gap while naturally supporting variable-length generation and conditioning on partial molecular context. However, balancing unconditional and context-conditioned generation remains challenging. We introduce KRONOS, a latent autoregressive diffusion framework that generates molecules in the latent space of a pre-trained autoencoder, jointly modeling molecular graph topology and geometry, while retaining the flexibility of autoregressive generation. We further introduce a mixed training strategy inspired by Fill-in-the Middle (FIM) paradigm, enabling both unconditional and fragment-conditioned molecular generation within a single left-to-right autoregressive model. Experiments on QM9 and GEOM-Drugs demonstrate that KRONOS achieves leading unconditional generation performance among autoregressive methods, while remaining competitive with diffusion models. Moreover, fragment-conditioned generation is achieved with negligible impact on unconditional generation performance, demonstrating that both generation paradigms can be supported within a single architecture.

Recent work has shown great progress in integrating spatial conditioning to control large, pre-trained text-to-image diffusion models. Despite these advances, existing methods describe the spatial image content using hand-crafted conditioning inputs, which are either semantically ambiguous (e.g., edges) or require expensive manual annotations (e.g., semantic segmentation). To address these limitations, we propose a new label-free way of conditioning diffusion models to enable fine-grained spatial control. We introduce the concept of neural semantic image synthesis, which uses neural layouts extracted from pre-trained foundation models as conditioning. Neural layouts are advantageous as they provide rich descriptions of the desired image, containing both semantics and detailed geometry of the scene. We experimentally show that images synthesized via neural semantic image synthesis achieve similar or superior pixel-level alignment of semantic classes compared to those created using expensive semantic label maps. At the same time, they capture better semantics, instance separation, and object orientation than other label-free conditioning options, such as edges or depth. Moreover, we show that images generated by neural layout conditioning can effectively augment real data for training various perception tasks.

Combining different models is a widely used paradigm in machine learning applications. While the most common approach is to form an ensemble of models and average their individual predictions, this approach is often rendered infeasible by given resource constraints in terms of memory and computation, which grow linearly with the number of models. We present a layer-wise model fusion algorithm for neural networks that utilizes optimal transport to (soft-) align neurons across the models before averaging their associated parameters. We show that this can successfully yield "one-shot" knowledge transfer (i.e, without requiring any retraining) between neural networks trained on heterogeneous non-i.i.d. data. In both i.i.d. and non-i.i.d. settings , we illustrate that our approach significantly outperforms vanilla averaging, as well as how it can serve as an efficient replacement for the ensemble with moderate fine-tuning, for standard convolutional networks (like VGG11), residual networks (like ResNet18), and multi-layer perceptrons on CIFAR10, CIFAR100, and MNIST. Finally, our approach also provides a principled way to combine the parameters of neural networks with different widths, and we explore its application for model compression. The code is available at the following link, https://github.com/sidak/otfusion.