Generative Flows

We describe Generative Blocks World to interact with the scene of a generated image by manipulating simple geometric abstractions. Our method represents scenes as assemblies of convex 3D primitives, and the same scene can be represented by different numbers of primitives, allowing an editor to move either whole structures or small details. Once the scene geometry has been edited, the image is generated by a flow-based method, which is conditioned on depth and a texture hint. Our texture hint takes into account the modified 3D primitives, exceeding the texture-consistency provided by existing techniques. These texture hints (a) allow accurate object and camera moves and (b) preserve the identity of objects. Our experiments demonstrate that our approach outperforms prior works in visual fidelity, editability, and compositional generalization. Code will be released.

Federated Learning (FL) is a privacy-preserving machine learning framework facilitating collaborative training across distributed clients. However, its performance is often compromised by data heterogeneity among participants, which can result in local models with limited generalization capability. Traditional model-homogeneous approaches address this issue primarily by regularizing local training procedures or dynamically adjusting client weights during aggregation. Nevertheless, these methods become unsuitable in scenarios involving clients with heterogeneous model architectures. In this paper, we propose a model-heterogeneous FL framework that enhances clients’ generalization performance on unseen data without relying on parameter aggregation. Instead of model parameters, clients share feature distribution statistics (mean and covariance) with the server. Then each client trains a variational transposed convolutional neural network using Gaussian latent variables sampled from these distributions, and use it to generate synthetic data. By fine-tuning local models with the synthetic data, clients achieve significant improvement of generalization ability. Experimental results demonstrate that our approach not only attains higher generalization accuracy compared to existing model-heterogeneous FL frameworks, but also reduces communication costs and memory consumption.

Flow-matching models deliver state-of-the-art fidelity in image and video generation, but the inherent sequential denoising process renders them slower. Existing acceleration methods like distillation, trajectory truncation, and consistency approaches are static, require retraining, and often fail to generalize across tasks. We propose FastFlow, a plug-and-play adaptive inference framework that accelerates generation in flow matching models. FastFlow identifies denoising steps that produce only minor adjustments to the denoising path and approximates them without using the full neural network models used for velocity predictions. The approximation utilizes finite-difference velocity estimates from prior predictions to efficiently extrapolate future states, enabling faster advancements along the denoising path at zero compute cost. This enables skipping computation at intermediary steps. We model the decision of how many steps to safely skip before requiring a full model computation as a multi-armed bandit problem. The bandit learns the optimal skips to balance speed with performance. FastFlow integrates seamlessly with existing pipelines and generalizes across image generation, video generation, and editing tasks. Experiments demonstrate a speedup of over $2.6\times$ while maintaining high-quality outputs. The source code for this work can be found at https://github.com/Div290/FastFlow.

Stochastic Interpolants (SI) are a powerful framework for generative modeling, capable of flexibly transforming between two probability distributions. However, their use in jointly optimized latent variable models remains unexplored as they require direct access to the samples from the two distributions. This work presents Latent Stochastic Interpolants (LSI) enabling joint learning in a latent space with end-to-end optimized encoder, decoder and latent SI models. We achieve this by developing a principled Evidence Lower Bound (ELBO) objective derived directly in continuous time. The joint optimization allows LSI to learn effective latent representations along with a generative process that transforms an arbitrary prior distribution into the encoder-defined aggregated posterior. LSI sidesteps the simple priors of the normal diffusion models and mitigates the computational demands of applying SI directly in high-dimensional observation spaces, while preserving the generative flexibility of the SI framework. We demonstrate the efficacy of LSI through comprehensive experiments on the standard large scale ImageNet generation benchmark.

Masked diffusion models (MDMs) have recently emerged as a promising alternative to autoregressive models over discrete domains. MDMs generate sequences in an any-order, parallel fashion, enabling fast inference and strong performance on non-causal tasks. However, a crucial limitation is that they do not support token insertions and are thus limited to *fixed-length* generations. To this end, we introduce **Flex**ible **M**asked **D**iffusion **M**odels (FlexMDMs), a discrete diffusion paradigm that simultaneously can model sequences of flexible length while provably retaining MDMs' flexibility of any-order inference. Grounded in an extension of the stochastic interpolant framework, FlexMDMs generate sequences by inserting mask tokens and unmasking them. Empirically, we show that FlexMDMs match MDMs in perplexity while modeling length statistics with much higher fidelity. On a synthetic maze planning task, they achieve $\approx$ 60\% higher success rate than MDM baselines. Finally, we show pretrained MDMs can easily be *retrofitted* into FlexMDMs: on 16 H100s, it takes only three days to fine-tune LLaDA-8B into a FlexMDM, achieving superior performance on math (GSM8K, 58\%$\to$67\%) and code infilling performance (52\%$\to$65\%).

Many proteins useful in modern medicine or bioengineering are challenging to make in the lab, fuse with other proteins in cells, or deliver to tissues in the body because their sequences are too long. Shortening these sequences typically involves costly, time-consuming experimental campaigns. Ideally, we could instead use modern models of massive databases of sequences from nature to learn how to propose shrunken proteins that resemble sequences found in nature. Unfortunately, these models struggle to efficiently search the combinatorial space of all deletions, and are not trained with inductive biases to learn how to delete. To address this gap, we propose SCISOR, a novel discrete diffusion model that deletes letters from sequences to generate protein samples that resemble those found in nature. To do so, SCISOR trains a de-noiser to reverse a forward noising process that adds random insertions to natural sequences. As a generative model, SCISOR fits evolutionary sequence data competitively with previous large models. In evaluation, SCISOR achieves state-of-the-art predictions of the functional effects of deletions on ProteinGym. Finally, we use the SCISOR de-noiser to shrink long protein sequences, and show that its suggested deletions result in significantly more realistic proteins and more often preserve functional motifs than previous models of evolutionary sequences.

Noise Contrastive Estimation (NCE) has fueled major breakthroughs in representation learning and generative modeling. Yet a long-standing challenge remains: accurately estimating ratios between distributions that differ substantially, which significantly limits the applicability of NCE on modern high-dimensional and multimodal datasets. We revisit this problem from a less explored perspective: the magnitude of the noise distribution. Specifically, we show that with a virtually scaled (i.e., artificially increased) noise magnitude, the gradient of the NCE objective can closely align with that of Maximum Likelihood, enabling a trajectory-wise approximation from NCE to MLE, and faster convergence both theoretically and empirically. Building on this insight, we introduce "Noisier" NCE, a simple drop-in modification to vanilla NCE that incurs little to no extra computational cost, while effectively handling density-ratio estimation in challenging regimes where traditional MLE and NCE struggle. Beyond improving classical density-ratio learning, "Noisier" NCE proves broadly applicable: it achieves strong results across image modeling, anomaly detection, and offline black-box optimization. On CIFAR-10 and ImageNet64×64 datasets, it yields 10-step and even 1-step samplers that match or surpass state-of-the-art methods, while cutting training iterations by up to half.

Deep generative models often face a fundamental tradeoff: high sample quality can come at the cost of memorisation, where the model reproduces training data rather than generalising across the underlying data geometry. We introduce Carré du champ flow matching (CDC-FM), a generalisation of flow matching (FM), that improves the quality-generalisation tradeoff by regularising the probability path with a geometry-aware noise. Our method replaces the homogeneous, isotropic noise in FM with a spatially varying, anisotropic Gaussian noise whose covariance captures the local geometry of the latent data manifold. We prove that this geometric noise can be optimally estimated from the data and is scalable to large data. Further, we provide an extensive experimental evaluation on diverse datasets (synthetic manifolds, point clouds, single-cell genomics, animal motion capture, and images) as well as various neural network architectures (MLPs, CNNs, and transformers). We demonstrate that CDC-FM consistently offers a better quality-generalisation tradeoff, even when used as a latent space generation model. We observe significant improvements over standard FM in data-scarce regimes and in highly non-uniformly sampled datasets, which are often encountered in AI for science applications. Our work provides a mathematical framework for studying the interplay between data geometry, generalisation and memorisation in generative models, as well as a robust and scalable algorithm that can be readily integrated into existing flow matching pipelines.

We propose Terminal Velocity Matching (TVM), a generalization of flow matching that enables high-fidelity one- and few-step generative modeling. TVM models the transition between any two diffusion timesteps and regularizes its behavior at its terminal time rather than at the initial time. We prove that TVM provides an upper bound on the $2$-Wasserstein distance between data and model distributions when the model is Lipschitz continuous. However, since Diffusion Transformers lack this property, we introduce minimal architectural changes that achieve stable, single-stage training. To make TVM efficient in practice, we develop a fused attention kernel that supports backward passes on Jacobian-Vector Products, which scale well with transformer architectures. On ImageNet-256x256, TVM achieves 3.29 FID with a single function evaluation (NFE) and 1.99 FID with 4 NFEs. It similarly achieves 4.32 1-NFE FID and 2.94 4-NFE FID on ImageNet-512x512, representing state-of-the-art performance for one/few-step models from scratch.

Diffusion models have achieved remarkable success in tasks ranging from image generation to inverse problems. However, training diffusion models typically requires clean ground-truth images, which are unavailable in many applications. We introduce the Measurement Score-based diffusion Model (MSM), a novel framework that learns partial measurement scores directly from noisy and subsampled measurements. By aggregating these scores in expectation, MSM synthesizes fully sampled measurements without requiring access to clean images. To make this practical, we develop a stochastic sampling variant of MSM that approximates the expectation efficiently and analyze its asymptotic equivalence to the exact formulation. We further extend MSM to posterior sampling for linear inverse problems, enabling accurate image reconstruction directly from partial scores. Experiments on natural images and multi-coil MRI demonstrate that MSM achieves state-of-the-art performance in unconditional generation and inverse problem solving---all while being trained exclusively on degraded measurements.

Flow-based Generative Models (FGMs) effectively transform noise into a data distribution, and coupling the noise and data in the training of FGM by Optimal Transport (OT) improves the straightness of the flow paths. However, existing OT- based couplings are difficult to combine with modern models and/or to scale to large datasets due to the curse of dimensionality in the sample complexity of (batch) OT. This paper introduces AlignFlow, a new approach using Semi-Discrete Optimal Transport (SDOT) to enhance FGM training by establishing explicit alignment between noise and data pairs. SDOT computes a transport map by partitioning the noise space into Laguerre cells, each mapped to a corresponding data point. During the training of FGM, i.i.d.-sampled noise is matched with corresponding data by the SDOT map. AlignFlow bypasses the curse of dimensionality and scales effectively to large datasets and models. Our experiments demonstrate that AlignFlow improves a wide range of state-of-the-art FGM algorithms and can be integrated as a plug-and-play solution with negligible additional cost.

Classifier-Free Guidance (CFG) is widely used to improve conditional fidelity in diffusion models, but its impact on sampling dynamics remains poorly understood. Prior studies, often restricted to unimodal conditional distributions or simplified cases, provide only a partial picture. We analyze CFG under multimodal conditionals and show that the sampling process unfolds in three successive stages. In the Direction Shift stage, guidance accelerates movement toward the weighted mean, introducing initialization bias and norm growth. In the Mode Separation stage, local dynamics remain largely neutral, but the inherited bias suppresses weaker modes, reducing global diversity. In the Concentration stage, guidance amplifies within-mode contraction, diminishing fine-grained variability. This unified view explains a widely observed phenomenon: stronger guidance improves semantic alignment but inevitably reduces diversity. Experiments support these predictions, showing that early strong guidance erodes global diversity, while late strong guidance suppresses fine-grained variation. Moreover, our theory naturally suggests a time-varying guidance schedule, and empirical results confirm that it consistently improves both quality and diversity.

Recent advances in large multi-modal generative models have demonstrated impressive capabilities in multi-modal generation, including image and video generation. These models are typically built upon multi-step frameworks like diffusion and flow matching, which inherently limits their inference efficiency (requiring 40-100 Number of Function Evaluations (NFEs)). While various few-step methods aim to accelerate the inference, existing solutions have clear limitations. Prominent distillation-based methods, such as progressive and consistency distillation, either require an iterative distillation procedure or show significant degradation at very few steps (< 4-NFE). Meanwhile, integrating adversarial training into distillation (e.g., DMD/DMD2 and SANA-Sprint) to enhance performance introduces training instability, added complexity, and high GPU memory overhead due to the auxiliary trained models. To this end, we propose TwinFlow, a simple yet effective framework for training 1-step generative models that bypasses the need for distillation from pre-trained models and avoids standard adversarial training, making it ideal for building large-scale, efficient models. On text-to-image tasks, our method achieves a GenEval score of 0.83 in 1-NFE, outperforming strong baselines like SANA-Sprint (a GAN loss-based framework) and RCGM (a consistency-based framework). **Notably, we demonstrate the scalability of TwinFlow by transforming Qwen-Image-20B---the current largest open-source multi-modal generative model---into an efficient few-step generator**. With just 1-NFE, our approach matches the performance of the original 100-NFE model on both the GenEval and DPG-Bench benchmarks, reducing computational cost by $100\times$ with minor quality degradation. Our code and models will be made publicly available.

Simulation-free training frameworks have been at the forefront of the generative modelling revolution in continuous spaces, leading to large-scale diffusion and flow matching models. However, such modern generative models suffer from expensive inference, inhibiting their use in numerous scientific applications like Boltzmann Generators (BGs) for molecular conformations that require fast likelihood evaluation. In this paper, we revisit classical normalizing flows in the context of BGs that offer efficient sampling and likelihoods, but whose training via maximum likelihood is often unstable and computationally challenging. We propose Regression Training of Normalizing Flows (RegFlow), a novel and scalable regression-based training objective that bypasses the numerical instability and computational challenge of conventional maximum likelihood training in favour of a simple $\ell_2$-regression objective. Specifically, RegFlow maps prior samples under our flow to targets computed using optimal transport couplings or a pre-trained continuous normalizing flow (CNF). To enhance numerical stability, RegFlow employs effective regularization strategies such as a new forward-backward self-consistency loss that enjoys painless implementation. Empirically, we demonstrate that RegFlow unlocks a broader class of architectures that were previously intractable to train for BGs with maximum likelihood. We also show RegFlow exceeds the performance, computational cost, and stability of maximum likelihood training in equilibrium sampling in Cartesian coordinates of alanine dipeptide, tripeptide, and tetrapeptide, showcasing its potential in molecular systems.

Various deep generative models have been proposed to estimate potential outcomes distributions from observational data. However, none of them have the favorable theoretical property of general Neyman-orthogonality and, associated with it, quasi-oracle efficiency and double robustness. In this paper, we introduce a general suite of generative Neyman-orthogonal (doubly-robust) learners that estimate the conditional distributions of potential outcomes. Our proposed generative doubly-robust learners (GDR-learners) are flexible and can be instantiated with many state-of-the-art deep generative models. In particular, we develop GDR-learners based on (a) conditional normalizing flows (which we call GDR-CNFs), (b) conditional generative adversarial networks (GDR-CGANs), (c) conditional variational autoencoders (GDR-CVAEs), and (d) conditional diffusion models (GDR-CDMs). Unlike the existing methods, our GDR-learners possess the properties of quasi-oracle efficiency and rate double robustness, and are thus asymptotically optimal. In a series of (semi-)synthetic experiments, we demonstrate that our GDR-learners are very effective and outperform the existing methods in estimating the conditional distributions of potential outcomes.

Probabilistic forecasting of joint distributions for irregular time series with missing values is an underexplored area in machine learning. Existing models, such as Gaussian Process Regression and ProFITi, are limited: while ProFITi is highly expressive due to its use of normalizing flows, it often produces unrealistic predictions because it lacks marginalization consistency—marginal distributions of subsets of variables may not match those predicted directly, leading to inaccurate marginal forecasts when trained on joints. We propose MOSES (Mixtures of Separable Flows), a novel model that parametrizes a stochastic process via a mixture of normalizing flows, where each component combines a latent multivariate Gaussian with separable univariate transformations. This design allows MOSES to be analytically marginalized, enabling accurate and reliable predictions for various probabilistic queries. Experiments on four datasets show that MOSES achieves highly accurate joint and marginal predictions. Thanks to its inherent marginalization consistency, MOSES significantly outperforms all baselines—including ProFITi—on marginal predictions. For joint predictions, it beats all other consistent models and performs close to or slightly worse than ProFITi.

Recent advancements in algorithms for local intrinsic dimension (LID) estimation have been closely tied to progress in neural networks (NN). However, NN architectures are often tailored to specific domains, such as audio or image data, incorporating inductive biases that limit their transferability across domains. Moreover, existing LID estimation methods leveraging these architectures are typically evaluated on either overly simplistic benchmarks or domain datasets where the true LID is unknown, resulting in potentially erroneous evaluations. To close this research gap, we first isolate problematic aspects of LID estimation and leverage them to analyze the limitations of state-of-the-art methods. Our approach employs several techniques to create LID benchmarks for arbitrary domains, including the introduction of a method to transform any manifold into the domain while preserving the manifold structure, thereby addressing challenges posed by biases in neural network-based methods. Our comparative analysis reveals critical limitations and identifies new directions for future development in LID estimation methods. Code will be available on github when published.

We present a novel framework for solving optimal transport (OT) problems based on the Hamilton--Jacobi (HJ) equation, whose viscosity solution uniquely characterizes the OT map. By leveraging the method of characteristics, we derive closed-form, bidirectional transport maps, thereby eliminating the need for numerical integration. The proposed method adopts a pure minimization framework: a single neural network is trained with a loss function derived from the method of characteristics of the HJ equation. This design guarantees convergence to the optimal map while eliminating adversarial training stages, thereby substantially reducing computational complexity. Furthermore, the framework naturally extends to a wide class of cost functions and supports class-conditional transport. Extensive experiments on diverse datasets demonstrate the accuracy, scalability, and efficiency of the proposed method, establishing it as a principled and versatile tool for OT applications with provable optimality.

Existing denoising generative models rely on solving discretized reverse-time SDEs or ODEs. In this paper, we identify a long-overlooked yet pervasive issue in this family of models: a misalignment between the pre-defined noise level and the actual noise level encoded in intermediate states during sampling. We refer to this misalignment as noise shift. Through empirical analysis, we demonstrate that noise shift is widespread in modern diffusion models and exhibits a systematic bias, leading to sub-optimal generation due to both out-of-distribution generalization and inaccurate denoising updates. To address this problem, we propose Noise Awareness Guidance (NAG), a simple yet effective correction method that explicitly steers sampling trajectories to remain consistent with the pre-defined noise schedule. We further introduce a classifier-free variant of NAG, which jointly trains a noise-conditional and a noise-unconditional model via noise-condition dropout, thereby eliminating the need for external classifiers. Extensive experiments, including ImageNet generation and various supervised fine-tuning tasks, show that NAG consistently mitigates noise shift and substantially improves the generation quality of mainstream diffusion models. Code will be released upon acceptance.

We present a novel theoretical framework for deep reinforcement learning (RL) in continuous environments by modeling the problem as a continuous-time stochastic process, drawing on insights from stochastic control. Building on previous work, we introduce a viable model of actor–critic algorithm that incorporates both exploration and stochastic transitions. For single-hidden-layer neural networks, we show that the state of the environment can be formulated as a two time scale process: the environment time and the gradient time. Within this formulation, we characterize how the time-dependent random variables that represent the environment's state and estimate of the cumulative discounted return evolve over gradient steps in the infinite width limit of two-layer networks. Using the theory of stochastic differential equations, we derive, for the first time in continuous RL, an equation describing the infinitesimal change in the state distribution at each gradient step, under a vanishingly small learning rate. Overall, our work provides a novel nonparametric formulation for studying overparametrized neural actor-critic algorithms. We empirically corroborate our theoretical result using a toy continuous control task.