Daily Papers

Recent advances in autoregressive (AR) models with continuous tokens for image generation show promising results by eliminating the need for discrete tokenization. However, these models face efficiency challenges due to their sequential token generation nature and reliance on computationally intensive diffusion-based sampling. We present ECAR (Efficient Continuous Auto-Regressive Image Generation via Multistage Modeling), an approach that addresses these limitations through two intertwined innovations: (1) a stage-wise continuous token generation strategy that reduces computational complexity and provides progressively refined token maps as hierarchical conditions, and (2) a multistage flow-based distribution modeling method that transforms only partial-denoised distributions at each stage comparing to complete denoising in normal diffusion models. Holistically, ECAR operates by generating tokens at increasing resolutions while simultaneously denoising the image at each stage. This design not only reduces token-to-image transformation cost by a factor of the stage number but also enables parallel processing at the token level. Our approach not only enhances computational efficiency but also aligns naturally with image generation principles by operating in continuous token space and following a hierarchical generation process from coarse to fine details. Experimental results demonstrate that ECAR achieves comparable image quality to DiT Peebles & Xie [2023] while requiring 10times FLOPs reduction and 5times speedup to generate a 256times256 image.

We formulate a hierarchical rectified flow to model data distributions. It hierarchically couples multiple ordinary differential equations (ODEs) and defines a time-differentiable stochastic process that generates a data distribution from a known source distribution. Each ODE resembles the ODE that is solved in a classic rectified flow, but differs in its domain, i.e., location, velocity, acceleration, etc. Unlike the classic rectified flow formulation, which formulates a single ODE in the location domain and only captures the expected velocity field (sufficient to capture a multi-modal data distribution), the hierarchical rectified flow formulation models the multi-modal random velocity field, acceleration field, etc., in their entirety. This more faithful modeling of the random velocity field enables integration paths to intersect when the underlying ODE is solved during data generation. Intersecting paths in turn lead to integration trajectories that are more straight than those obtained in the classic rectified flow formulation, where integration paths cannot intersect. This leads to modeling of data distributions with fewer neural function evaluations. We empirically verify this on synthetic 1D and 2D data as well as MNIST, CIFAR-10, and ImageNet-32 data. Our code is available at: https://riccizz.github.io/HRF/.

Large-scale commercial platforms usually involve numerous business domains for diverse business strategies and expect their recommendation systems to provide click-through rate (CTR) predictions for multiple domains simultaneously. Existing promising and widely-used multi-domain models discover domain relationships by explicitly constructing domain-specific networks, but the computation and memory boost significantly with the increase of domains. To reduce computational complexity, manually grouping domains with particular business strategies is common in industrial applications. However, this pre-defined data partitioning way heavily relies on prior knowledge, and it may neglect the underlying data distribution of each domain, hence limiting the model's representation capability. Regarding the above issues, we propose an elegant and flexible multi-distribution modeling paradigm, named Adaptive Distribution Hierarchical Model (AdaptDHM), which is an end-to-end optimization hierarchical structure consisting of a clustering process and classification process. Specifically, we design a distribution adaptation module with a customized dynamic routing mechanism. Instead of introducing prior knowledge for pre-defined data allocation, this routing algorithm adaptively provides a distribution coefficient for each sample to determine which cluster it belongs to. Each cluster corresponds to a particular distribution so that the model can sufficiently capture the commonalities and distinctions between these distinct clusters. Extensive experiments on both public and large-scale Alibaba industrial datasets verify the effectiveness and efficiency of AdaptDHM: Our model achieves impressive prediction accuracy and its time cost during the training stage is more than 50% less than that of other models.

In reinforcement learning and imitation learning, an object of central importance is the state distribution induced by the policy. It plays a crucial role in the policy gradient theorem, and references to it--along with the related state-action distribution--can be found all across the literature. Despite its importance, the state distribution is mostly discussed indirectly and theoretically, rather than being modeled explicitly. The reason being an absence of appropriate density estimation tools. In this work, we investigate applications of a normalizing flow-based model for the aforementioned distributions. In particular, we use a pair of flows coupled through the optimality point of the Donsker-Varadhan representation of the Kullback-Leibler (KL) divergence, for distribution matching based imitation learning. Our algorithm, Coupled Flow Imitation Learning (CFIL), achieves state-of-the-art performance on benchmark tasks with a single expert trajectory and extends naturally to a variety of other settings, including the subsampled and state-only regimes.

Learning the tail behavior of a distribution is a notoriously difficult problem. By definition, the number of samples from the tail is small, and deep generative models, such as normalizing flows, tend to concentrate on learning the body of the distribution. In this paper, we focus on improving the ability of normalizing flows to correctly capture the tail behavior and, thus, form more accurate models. We prove that the marginal tailedness of an autoregressive flow can be controlled via the tailedness of the marginals of its base distribution. This theoretical insight leads us to a novel type of flows based on flexible base distributions and data-driven linear layers. An empirical analysis shows that the proposed method improves on the accuracy -- especially on the tails of the distribution -- and is able to generate heavy-tailed data. We demonstrate its application on a weather and climate example, in which capturing the tail behavior is essential.

Conventional diffusion models typically relies on a fixed forward process, which implicitly defines complex marginal distributions over latent variables. This can often complicate the reverse process' task in learning generative trajectories, and results in costly inference for diffusion models. To address these limitations, we introduce Neural Flow Diffusion Models (NFDM), a novel framework that enhances diffusion models by supporting a broader range of forward processes beyond the fixed linear Gaussian. We also propose a novel parameterization technique for learning the forward process. Our framework provides an end-to-end, simulation-free optimization objective, effectively minimizing a variational upper bound on the negative log-likelihood. Experimental results demonstrate NFDM's strong performance, evidenced by state-of-the-art likelihood estimation. Furthermore, we investigate NFDM's capacity for learning generative dynamics with specific characteristics, such as deterministic straight lines trajectories. This exploration underscores NFDM's versatility and its potential for a wide range of applications.

Flow-based generative models are powerful exact likelihood models with efficient sampling and inference. Despite their computational efficiency, flow-based models generally have much worse density modeling performance compared to state-of-the-art autoregressive models. In this paper, we investigate and improve upon three limiting design choices employed by flow-based models in prior work: the use of uniform noise for dequantization, the use of inexpressive affine flows, and the use of purely convolutional conditioning networks in coupling layers. Based on our findings, we propose Flow++, a new flow-based model that is now the state-of-the-art non-autoregressive model for unconditional density estimation on standard image benchmarks. Our work has begun to close the significant performance gap that has so far existed between autoregressive models and flow-based models. Our implementation is available at https://github.com/aravindsrinivas/flowpp

We present Piecewise Rectified Flow (PeRFlow), a flow-based method for accelerating diffusion models. PeRFlow divides the sampling process of generative flows into several time windows and straightens the trajectories in each interval via the reflow operation, thereby approaching piecewise linear flows. PeRFlow achieves superior performance in a few-step generation. Moreover, through dedicated parameterizations, the obtained PeRFlow models show advantageous transfer ability, serving as universal plug-and-play accelerators that are compatible with various workflows based on the pre-trained diffusion models. The implementations of training and inference are fully open-sourced. https://github.com/magic-research/piecewise-rectified-flow

Generating novel molecules with higher properties than the training space, namely the out-of-distribution generation, is important for {de~novo} drug design. However, it is not easy for distribution learning-based models, for example diffusion models, to solve this challenge as these methods are designed to fit the distribution of training data as close as possible. In this paper, we show that Bayesian flow network is capable of effortlessly generating high quality out-of-distribution samples that meet several scenarios. We introduce a semi-autoregressive training/sampling method that helps to enhance the model performance and surpass the state-of-the-art models.

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.

Diffusion models have emerged as a promising alternative to autoregressive models in modeling discrete categorical data. Yet diffusion models that directly work on discrete data space do not fully exploit the power of iterative refinement, as the signals are lost during the transition between discrete states. Existing continuous diffusion models for discrete data have limited performance compared to discrete approaches, and the unclear link between them restricts the development of diffusion models for discrete data. In this work, we propose a continuous diffusion model for language modeling that incorporates the geometry of the underlying categorical distribution. We establish a connection between the discrete diffusion and continuous flow on the statistical manifold, and building on the analogy, we introduce a simple design for the diffusion process that generalizes previous discrete diffusion models. We further propose a simulation-free training framework based on radial symmetry and a simple technique to address the high dimensionality of the manifold. Comprehensive experiments on language modeling benchmarks and other modalities show that our method outperforms existing discrete diffusion models and approaches the performance of autoregressive models. Codes available at https://github.com/harryjo97/RDLM{https://github.com/harryjo97/RDLM}.

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.

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.

Numerous biological and physical processes can be modeled as systems of interacting entities evolving continuously over time, e.g. the dynamics of communicating cells or physical particles. Learning the dynamics of such systems is essential for predicting the temporal evolution of populations across novel samples and unseen environments. Flow-based models allow for learning these dynamics at the population level - they model the evolution of the entire distribution of samples. However, current flow-based models are limited to a single initial population and a set of predefined conditions which describe different dynamics. We argue that multiple processes in natural sciences have to be represented as vector fields on the Wasserstein manifold of probability densities. That is, the change of the population at any moment in time depends on the population itself due to the interactions between samples. In particular, this is crucial for personalized medicine where the development of diseases and their respective treatment response depends on the microenvironment of cells specific to each patient. We propose Meta Flow Matching (MFM), a practical approach to integrating along these vector fields on the Wasserstein manifold by amortizing the flow model over the initial populations. Namely, we embed the population of samples using a Graph Neural Network (GNN) and use these embeddings to train a Flow Matching model. This gives MFM the ability to generalize over the initial distributions unlike previously proposed methods. We demonstrate the ability of MFM to improve prediction of individual treatment responses on a large scale multi-patient single-cell drug screen dataset.

Normalising Flows are non-parametric statistical models characterised by their dual capabilities of density estimation and generation. This duality requires an inherently invertible architecture. However, the requirement of invertibility imposes constraints on their expressiveness, necessitating a large number of parameters and innovative architectural designs to achieve good results. Whilst flow-based models predominantly rely on neural-network-based transformations for expressive designs, alternative transformation methods have received limited attention. In this work, we present Ferumal flow, a novel kernelised normalising flow paradigm that integrates kernels into the framework. Our results demonstrate that a kernelised flow can yield competitive or superior results compared to neural network-based flows whilst maintaining parameter efficiency. Kernelised flows excel especially in the low-data regime, enabling flexible non-parametric density estimation in applications with sparse data availability.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

Diffusion models excel in high-quality generation but suffer from slow inference due to iterative sampling. While recent methods have successfully transformed diffusion models into one-step generators, they neglect model size reduction, limiting their applicability in compute-constrained scenarios. This paper aims to develop small, efficient one-step diffusion models based on the powerful rectified flow framework, by exploring joint compression of inference steps and model size. The rectified flow framework trains one-step generative models using two operations, reflow and distillation. Compared with the original framework, squeezing the model size brings two new challenges: (1) the initialization mismatch between large teachers and small students during reflow; (2) the underperformance of naive distillation on small student models. To overcome these issues, we propose Annealing Reflow and Flow-Guided Distillation, which together comprise our SlimFlow framework. With our novel framework, we train a one-step diffusion model with an FID of 5.02 and 15.7M parameters, outperforming the previous state-of-the-art one-step diffusion model (FID=6.47, 19.4M parameters) on CIFAR10. On ImageNet 64times64 and FFHQ 64times64, our method yields small one-step diffusion models that are comparable to larger models, showcasing the effectiveness of our method in creating compact, efficient one-step diffusion models.

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.

Despite Flow Matching and diffusion models having emerged as powerful generative paradigms for continuous variables such as images and videos, their application to high-dimensional discrete data, such as language, is still limited. In this work, we present Discrete Flow Matching, a novel discrete flow paradigm designed specifically for generating discrete data. Discrete Flow Matching offers several key contributions: (i) it works with a general family of probability paths interpolating between source and target distributions; (ii) it allows for a generic formula for sampling from these probability paths using learned posteriors such as the probability denoiser (x-prediction) and noise-prediction (epsilon-prediction); (iii) practically, focusing on specific probability paths defined with different schedulers considerably improves generative perplexity compared to previous discrete diffusion and flow models; and (iv) by scaling Discrete Flow Matching models up to 1.7B parameters, we reach 6.7% Pass@1 and 13.4% Pass@10 on HumanEval and 6.7% Pass@1 and 20.6% Pass@10 on 1-shot MBPP coding benchmarks. Our approach is capable of generating high-quality discrete data in a non-autoregressive fashion, significantly closing the gap between autoregressive models and discrete flow models.

Diffusion-based generative models (DBGMs) perturb data to a target noise distribution and reverse this process to generate samples. The choice of noising process, or inference diffusion process, affects both likelihoods and sample quality. For example, extending the inference process with auxiliary variables leads to improved sample quality. While there are many such multivariate diffusions to explore, each new one requires significant model-specific analysis, hindering rapid prototyping and evaluation. In this work, we study Multivariate Diffusion Models (MDMs). For any number of auxiliary variables, we provide a recipe for maximizing a lower-bound on the MDMs likelihood without requiring any model-specific analysis. We then demonstrate how to parameterize the diffusion for a specified target noise distribution; these two points together enable optimizing the inference diffusion process. Optimizing the diffusion expands easy experimentation from just a few well-known processes to an automatic search over all linear diffusions. To demonstrate these ideas, we introduce two new specific diffusions as well as learn a diffusion process on the MNIST, CIFAR10, and ImageNet32 datasets. We show learned MDMs match or surpass bits-per-dims (BPDs) relative to fixed choices of diffusions for a given dataset and model architecture.

We tackle the problem of sampling from intractable high-dimensional density functions, a fundamental task that often appears in machine learning and statistics. We extend recent sampling-based approaches that leverage controlled stochastic processes to model approximate samples from these target densities. The main drawback of these approaches is that the training objective requires full trajectories to compute, resulting in sluggish credit assignment issues due to use of entire trajectories and a learning signal present only at the terminal time. In this work, we present Diffusion Generative Flow Samplers (DGFS), a sampling-based framework where the learning process can be tractably broken down into short partial trajectory segments, via parameterizing an additional "flow function". Our method takes inspiration from the theory developed for generative flow networks (GFlowNets), allowing us to make use of intermediate learning signals. Through various challenging experiments, we demonstrate that DGFS achieves more accurate estimates of the normalization constant than closely-related prior methods.

The choice of approximate posterior distribution is one of the core problems in variational inference. Most applications of variational inference employ simple families of posterior approximations in order to allow for efficient inference, focusing on mean-field or other simple structured approximations. This restriction has a significant impact on the quality of inferences made using variational methods. We introduce a new approach for specifying flexible, arbitrarily complex and scalable approximate posterior distributions. Our approximations are distributions constructed through a normalizing flow, whereby a simple initial density is transformed into a more complex one by applying a sequence of invertible transformations until a desired level of complexity is attained. We use this view of normalizing flows to develop categories of finite and infinitesimal flows and provide a unified view of approaches for constructing rich posterior approximations. We demonstrate that the theoretical advantages of having posteriors that better match the true posterior, combined with the scalability of amortized variational approaches, provides a clear improvement in performance and applicability of variational inference.

Generative flow networks (GFlowNets) are amortized variational inference algorithms that are trained to sample from unnormalized target distributions over compositional objects. A key limitation of GFlowNets until this time has been that they are restricted to discrete spaces. We present a theory for generalized GFlowNets, which encompasses both existing discrete GFlowNets and ones with continuous or hybrid state spaces, and perform experiments with two goals in mind. First, we illustrate critical points of the theory and the importance of various assumptions. Second, we empirically demonstrate how observations about discrete GFlowNets transfer to the continuous case and show strong results compared to non-GFlowNet baselines on several previously studied tasks. This work greatly widens the perspectives for the application of GFlowNets in probabilistic inference and various modeling settings.

We propose an inference-time scaling approach for pretrained flow models. Recently, inference-time scaling has gained significant attention in LLMs and diffusion models, improving sample quality or better aligning outputs with user preferences by leveraging additional computation. For diffusion models, particle sampling has allowed more efficient scaling due to the stochasticity at intermediate denoising steps. On the contrary, while flow models have gained popularity as an alternative to diffusion models--offering faster generation and high-quality outputs in state-of-the-art image and video generative models--efficient inference-time scaling methods used for diffusion models cannot be directly applied due to their deterministic generative process. To enable efficient inference-time scaling for flow models, we propose three key ideas: 1) SDE-based generation, enabling particle sampling in flow models, 2) Interpolant conversion, broadening the search space and enhancing sample diversity, and 3) Rollover Budget Forcing (RBF), an adaptive allocation of computational resources across timesteps to maximize budget utilization. Our experiments show that SDE-based generation, particularly variance-preserving (VP) interpolant-based generation, improves the performance of particle sampling methods for inference-time scaling in flow models. Additionally, we demonstrate that RBF with VP-SDE achieves the best performance, outperforming all previous inference-time scaling approaches.

We incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2021) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes.

This paper introduces Bayesian Flow Networks (BFNs), a new class of generative model in which the parameters of a set of independent distributions are modified with Bayesian inference in the light of noisy data samples, then passed as input to a neural network that outputs a second, interdependent distribution. Starting from a simple prior and iteratively updating the two distributions yields a generative procedure similar to the reverse process of diffusion models; however it is conceptually simpler in that no forward process is required. Discrete and continuous-time loss functions are derived for continuous, discretised and discrete data, along with sample generation procedures. Notably, the network inputs for discrete data lie on the probability simplex, and are therefore natively differentiable, paving the way for gradient-based sample guidance and few-step generation in discrete domains such as language modelling. The loss function directly optimises data compression and places no restrictions on the network architecture. In our experiments BFNs achieve competitive log-likelihoods for image modelling on dynamically binarized MNIST and CIFAR-10, and outperform all known discrete diffusion models on the text8 character-level language modelling task.

Normalizing Flows (NFs) are likelihood-based models for continuous inputs. They have demonstrated promising results on both density estimation and generative modeling tasks, but have received relatively little attention in recent years. In this work, we demonstrate that NFs are more powerful than previously believed. We present TarFlow: a simple and scalable architecture that enables highly performant NF models. TarFlow can be thought of as a Transformer-based variant of Masked Autoregressive Flows (MAFs): it consists of a stack of autoregressive Transformer blocks on image patches, alternating the autoregression direction between layers. TarFlow is straightforward to train end-to-end, and capable of directly modeling and generating pixels. We also propose three key techniques to improve sample quality: Gaussian noise augmentation during training, a post training denoising procedure, and an effective guidance method for both class-conditional and unconditional settings. Putting these together, TarFlow sets new state-of-the-art results on likelihood estimation for images, beating the previous best methods by a large margin, and generates samples with quality and diversity comparable to diffusion models, for the first time with a stand-alone NF model. We make our code available at https://github.com/apple/ml-tarflow.

The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) -- an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.

We present a new algorithm to optimize distributions defined implicitly by parameterized stochastic diffusions. Doing so allows us to modify the outcome distribution of sampling processes by optimizing over their parameters. We introduce a general framework for first-order optimization of these processes, that performs jointly, in a single loop, optimization and sampling steps. This approach is inspired by recent advances in bilevel optimization and automatic implicit differentiation, leveraging the point of view of sampling as optimization over the space of probability distributions. We provide theoretical guarantees on the performance of our method, as well as experimental results demonstrating its effectiveness in real-world settings.

We propose a principled and effective framework for one-step generative modeling. We introduce the notion of average velocity to characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. A well-defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the MeanFlow model, is self-contained and requires no pre-training, distillation, or curriculum learning. MeanFlow demonstrates strong empirical performance: it achieves an FID of 3.43 with a single function evaluation (1-NFE) on ImageNet 256x256 trained from scratch, significantly outperforming previous state-of-the-art one-step diffusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will motivate future research to revisit the foundations of these powerful models.

Normalizing flows (NFs) have become a prominent method for deep generative models that allow for an analytic probability density estimation and efficient synthesis. However, a flow-based network is considered to be inefficient in parameter complexity because of reduced expressiveness of bijective mapping, which renders the models unfeasibly expensive in terms of parameters. We present an alternative parameterization scheme called NanoFlow, which uses a single neural density estimator to model multiple transformation stages. Hence, we propose an efficient parameter decomposition method and the concept of flow indication embedding, which are key missing components that enable density estimation from a single neural network. Experiments performed on audio and image models confirm that our method provides a new parameter-efficient solution for scalable NFs with significant sublinear parameter complexity.

This work presents mixed variational flows (MixFlows), a new variational family that consists of a mixture of repeated applications of a map to an initial reference distribution. First, we provide efficient algorithms for i.i.d. sampling, density evaluation, and unbiased ELBO estimation. We then show that MixFlows have MCMC-like convergence guarantees when the flow map is ergodic and measure-preserving, and provide bounds on the accumulation of error for practical implementations where the flow map is approximated. Finally, we develop an implementation of MixFlows based on uncorrected discretized Hamiltonian dynamics combined with deterministic momentum refreshment. Simulated and real data experiments show that MixFlows can provide more reliable posterior approximations than several black-box normalizing flows, as well as samples of comparable quality to those obtained from state-of-the-art MCMC methods.

Conventional wisdom holds that autoregressive models for image generation are typically accompanied by vector-quantized tokens. We observe that while a discrete-valued space can facilitate representing a categorical distribution, it is not a necessity for autoregressive modeling. In this work, we propose to model the per-token probability distribution using a diffusion procedure, which allows us to apply autoregressive models in a continuous-valued space. Rather than using categorical cross-entropy loss, we define a Diffusion Loss function to model the per-token probability. This approach eliminates the need for discrete-valued tokenizers. We evaluate its effectiveness across a wide range of cases, including standard autoregressive models and generalized masked autoregressive (MAR) variants. By removing vector quantization, our image generator achieves strong results while enjoying the speed advantage of sequence modeling. We hope this work will motivate the use of autoregressive generation in other continuous-valued domains and applications.

While previous distribution shift detection approaches can identify if a shift has occurred, these approaches cannot localize which specific features have caused a distribution shift -- a critical step in diagnosing or fixing any underlying issue. For example, in military sensor networks, users will want to detect when one or more of the sensors has been compromised, and critically, they will want to know which specific sensors might be compromised. Thus, we first define a formalization of this problem as multiple conditional distribution hypothesis tests and propose both non-parametric and parametric statistical tests. For both efficiency and flexibility, we then propose to use a test statistic based on the density model score function (i.e. gradient with respect to the input) -- which can easily compute test statistics for all dimensions in a single forward and backward pass. Any density model could be used for computing the necessary statistics including deep density models such as normalizing flows or autoregressive models. We additionally develop methods for identifying when and where a shift occurs in multivariate time-series data and show results for multiple scenarios using realistic attack models on both simulated and real world data.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

Simulation-free methods for training continuous-time generative models construct probability paths that go between noise distributions and individual data samples. Recent works, such as Flow Matching, derived paths that are optimal for each data sample. However, these algorithms rely on independent data and noise samples, and do not exploit underlying structure in the data distribution for constructing probability paths. We propose Multisample Flow Matching, a more general framework that uses non-trivial couplings between data and noise samples while satisfying the correct marginal constraints. At very small overhead costs, this generalization allows us to (i) reduce gradient variance during training, (ii) obtain straighter flows for the learned vector field, which allows us to generate high-quality samples using fewer function evaluations, and (iii) obtain transport maps with lower cost in high dimensions, which has applications beyond generative modeling. Importantly, we do so in a completely simulation-free manner with a simple minimization objective. We show that our proposed methods improve sample consistency on downsampled ImageNet data sets, and lead to better low-cost sample generation.

The Distribution Matching Distillation (DMD) has been successfully applied to text-to-image diffusion models such as Stable Diffusion (SD) 1.5. However, vanilla DMD suffers from convergence difficulties on large-scale flow-based text-to-image models, such as SD 3.5 and FLUX. In this paper, we first analyze the issues when applying vanilla DMD on large-scale models. Then, to overcome the scalability challenge, we propose implicit distribution alignment (IDA) to regularize the distance between the generator and fake distribution. Furthermore, we propose intra-segment guidance (ISG) to relocate the timestep importance distribution from the teacher model. With IDA alone, DMD converges for SD 3.5; employing both IDA and ISG, DMD converges for SD 3.5 and FLUX.1 dev. Along with other improvements such as scaled up discriminator models, our final model, dubbed SenseFlow, achieves superior performance in distillation for both diffusion based text-to-image models such as SDXL, and flow-matching models such as SD 3.5 Large and FLUX. The source code will be avaliable at https://github.com/XingtongGe/SenseFlow.

Flow matching (FM) is a general framework for defining probability paths via Ordinary Differential Equations (ODEs) to transform between noise and data samples. Recent approaches attempt to straighten these flow trajectories to generate high-quality samples with fewer function evaluations, typically through iterative rectification methods or optimal transport solutions. In this paper, we introduce Consistency Flow Matching (Consistency-FM), a novel FM method that explicitly enforces self-consistency in the velocity field. Consistency-FM directly defines straight flows starting from different times to the same endpoint, imposing constraints on their velocity values. Additionally, we propose a multi-segment training approach for Consistency-FM to enhance expressiveness, achieving a better trade-off between sampling quality and speed. Preliminary experiments demonstrate that our Consistency-FM significantly improves training efficiency by converging 4.4x faster than consistency models and 1.7x faster than rectified flow models while achieving better generation quality. Our code is available at: https://github.com/YangLing0818/consistency_flow_matching

Flow-based generative models parameterize probability distributions through an invertible transformation and can be trained by maximum likelihood. Invertible residual networks provide a flexible family of transformations where only Lipschitz conditions rather than strict architectural constraints are needed for enforcing invertibility. However, prior work trained invertible residual networks for density estimation by relying on biased log-density estimates whose bias increased with the network's expressiveness. We give a tractable unbiased estimate of the log density using a "Russian roulette" estimator, and reduce the memory required during training by using an alternative infinite series for the gradient. Furthermore, we improve invertible residual blocks by proposing the use of activation functions that avoid derivative saturation and generalizing the Lipschitz condition to induced mixed norms. The resulting approach, called Residual Flows, achieves state-of-the-art performance on density estimation amongst flow-based models, and outperforms networks that use coupling blocks at joint generative and discriminative modeling.

Most real-world graphs exhibit a hierarchical structure, which is often overlooked by existing graph generation methods. To address this limitation, we propose a novel graph generative network that captures the hierarchical nature of graphs and successively generates the graph sub-structures in a coarse-to-fine fashion. At each level of hierarchy, this model generates communities in parallel, followed by the prediction of cross-edges between communities using separate neural networks. This modular approach enables scalable graph generation for large and complex graphs. Moreover, we model the output distribution of edges in the hierarchical graph with a multinomial distribution and derive a recursive factorization for this distribution. This enables us to generate community graphs with integer-valued edge weights in an autoregressive manner. Empirical studies demonstrate the effectiveness and scalability of our proposed generative model, achieving state-of-the-art performance in terms of graph quality across various benchmark datasets. The code is available at https://github.com/Karami-m/HiGen_main.

A normalizing flow models a complex probability density as an invertible transformation of a simple base density. Flows based on either coupling or autoregressive transforms both offer exact density evaluation and sampling, but rely on the parameterization of an easily invertible elementwise transformation, whose choice determines the flexibility of these models. Building upon recent work, we propose a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of both coupling and autoregressive transforms while retaining analytic invertibility. We demonstrate that neural spline flows improve density estimation, variational inference, and generative modeling of images.

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N{+}D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D{=}1 and to diffusion models when D{to}infty. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D{to} infty) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64{times}64 datasets, with FID scores of 1.91/2.43 when D{=}2048/128. In class-conditional setting, D{=}2048 yields current state-of-the-art FID of 1.74 on CIFAR-10. In addition, we demonstrate that models with smaller D exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

Mixed Integer Linear Programming (MILP) is a fundamental tool for modeling combinatorial optimization problems. Recently, a growing body of research has used machine learning to accelerate MILP solving. Despite the increasing popularity of this approach, there is a lack of a common repository that provides distributions of similar MILP instances across different domains, at different hardness levels, with standardized test sets. In this paper, we introduce Distributional MIPLIB, a multi-domain library of problem distributions for advancing ML-guided MILP methods. We curate MILP distributions from existing work in this area as well as real-world problems that have not been used, and classify them into different hardness levels. It will facilitate research in this area by enabling comprehensive evaluation on diverse and realistic domains. We empirically illustrate the benefits of using Distributional MIPLIB as a research vehicle in two ways. We evaluate the performance of ML-guided variable branching on previously unused distributions to identify potential areas for improvement. Moreover, we propose to learn branching policies from a mix of distributions, demonstrating that mixed distributions achieve better performance compared to homogeneous distributions when there is limited data and generalize well to larger instances. The dataset is publicly available at https://sites.google.com/usc.edu/distributional-miplib/home.

We introduce Autoregressive Diffusion Models (ARDMs), a model class encompassing and generalizing order-agnostic autoregressive models (Uria et al., 2014) and absorbing discrete diffusion (Austin et al., 2021), which we show are special cases of ARDMs under mild assumptions. ARDMs are simple to implement and easy to train. Unlike standard ARMs, they do not require causal masking of model representations, and can be trained using an efficient objective similar to modern probabilistic diffusion models that scales favourably to highly-dimensional data. At test time, ARDMs support parallel generation which can be adapted to fit any given generation budget. We find that ARDMs require significantly fewer steps than discrete diffusion models to attain the same performance. Finally, we apply ARDMs to lossless compression, and show that they are uniquely suited to this task. Contrary to existing approaches based on bits-back coding, ARDMs obtain compelling results not only on complete datasets, but also on compressing single data points. Moreover, this can be done using a modest number of network calls for (de)compression due to the model's adaptable parallel generation.

Solving Bayesian inverse problems efficiently remains a significant challenge due to the complexity of posterior distributions and the computational cost of traditional sampling methods. Given a series of observations and the forward model, we want to recover the distribution of the parameters, conditioned on observed experimental data. We show, that combining Conditional Flow Mathching (CFM) with transformer-based architecture, we can efficiently sample from such kind of distribution, conditioned on variable number of observations.

Consistency models (CMs) are a powerful class of diffusion-based generative models optimized for fast sampling. Most existing CMs are trained using discretized timesteps, which introduce additional hyperparameters and are prone to discretization errors. While continuous-time formulations can mitigate these issues, their success has been limited by training instability. To address this, we propose a simplified theoretical framework that unifies previous parameterizations of diffusion models and CMs, identifying the root causes of instability. Based on this analysis, we introduce key improvements in diffusion process parameterization, network architecture, and training objectives. These changes enable us to train continuous-time CMs at an unprecedented scale, reaching 1.5B parameters on ImageNet 512x512. Our proposed training algorithm, using only two sampling steps, achieves FID scores of 2.06 on CIFAR-10, 1.48 on ImageNet 64x64, and 1.88 on ImageNet 512x512, narrowing the gap in FID scores with the best existing diffusion models to within 10%.

Modeling and estimating mixed memberships for overlapping unipartite un-weighted networks has been well studied in recent years. However, to our knowledge, there is no model for a more general case, the overlapping bipartite weighted networks. To close this gap, we introduce a novel model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. Our model allows an adjacency matrix to follow any distribution as long as its expectation has a block structure related to node membership. In particular, BiMMDF can model overlapping bipartite signed networks and it is an extension of many previous models, including the popular mixed membership stochastic blcokmodels. An efficient algorithm with a theoretical guarantee of consistent estimation is applied to fit BiMMDF. We then obtain the separation conditions of BiMMDF for different distributions. Furthermore, we also consider missing edges for sparse networks. The advantage of BiMMDF is demonstrated in extensive synthetic networks and eight real-world networks.

Diffusion (score-based) generative models have been widely used for modeling various types of complex data, including images, audios, and point clouds. Recently, the deep connection between forward-backward stochastic differential equations (SDEs) and diffusion-based models has been revealed, and several new variants of SDEs are proposed (e.g., sub-VP, critically-damped Langevin) along this line. Despite the empirical success of the hand-crafted fixed forward SDEs, a great quantity of proper forward SDEs remain unexplored. In this work, we propose a general framework for parameterizing the diffusion model, especially the spatial part of the forward SDE. An abstract formalism is introduced with theoretical guarantees, and its connection with previous diffusion models is leveraged. We demonstrate the theoretical advantage of our method from an optimization perspective. Numerical experiments on synthetic datasets, MINIST and CIFAR10 are also presented to validate the effectiveness of our framework.

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

Diffusion- and flow-based models have emerged as state-of-the-art generative modeling approaches, but they require many sampling steps. Consistency models can distill these models into efficient one-step generators; however, unlike flow- and diffusion-based methods, their performance inevitably degrades when increasing the number of steps, which we show both analytically and empirically. Flow maps generalize these approaches by connecting any two noise levels in a single step and remain effective across all step counts. In this paper, we introduce two new continuous-time objectives for training flow maps, along with additional novel training techniques, generalizing existing consistency and flow matching objectives. We further demonstrate that autoguidance can improve performance, using a low-quality model for guidance during distillation, and an additional boost can be achieved by adversarial finetuning, with minimal loss in sample diversity. We extensively validate our flow map models, called Align Your Flow, on challenging image generation benchmarks and achieve state-of-the-art few-step generation performance on both ImageNet 64x64 and 512x512, using small and efficient neural networks. Finally, we show text-to-image flow map models that outperform all existing non-adversarially trained few-step samplers in text-conditioned synthesis.

We explore a generative machine learning-based approach for estimating multi-dimensional probability density functions (PDFs) in a target sample using a statistically independent but related control sample - a common challenge in particle physics data analysis. The generative model must accurately reproduce individual observable distributions while preserving the correlations between them, based on the input multidimensional distribution from the control sample. Here we present a conditional normalizing flow model (CNF) based on a chain of bijectors which learns to transform unpaired simulation events to data events. We assess the performance of the CNF model in the context of LHC Higgs to diphoton analysis, where we use the CNF model to convert a Monte Carlo diphoton sample to one that models data. We show that the CNF model can accurately model complex data distributions and correlations. We also leverage the recently popularized Modified Differential Multiplier Method (MDMM) to improve the convergence of our model and assign physical meaning to usually arbitrary loss-function parameters.

Generative machine learning methods, such as diffusion models and flow matching, have shown great potential in modeling complex system behaviors and building efficient surrogate models. However, these methods typically learn the underlying physics implicitly from data. We propose Physics-Based Flow Matching (PBFM), a novel generative framework that explicitly embeds physical constraints, both PDE residuals and algebraic relations, into the flow matching objective. We also introduce temporal unrolling at training time that improves the accuracy of the final, noise-free sample prediction. Our method jointly minimizes the flow matching loss and the physics-based residual loss without requiring hyperparameter tuning of their relative weights. Additionally, we analyze the role of the minimum noise level, sigma_{min}, in the context of physical constraints and evaluate a stochastic sampling strategy that helps to reduce physical residuals. Through extensive benchmarks on three representative PDE problems, we show that our approach yields up to an 8times more accurate physical residuals compared to FM, while clearly outperforming existing algorithms in terms of distributional accuracy. PBFM thus provides a principled and efficient framework for surrogate modeling, uncertainty quantification, and accelerated simulation in physics and engineering applications.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

Large Language Models (LLMs) are increasingly being used to simulate human-like decision making in agent-based financial market models (ABMs). As models become more powerful and accessible, researchers can now incorporate individual LLM decisions into ABM environments. However, integration may introduce inherent biases that need careful evaluation. In this paper we test three state-of-the-art GPT models for bias using two model sampling approaches: one-shot and few-shot API queries. We observe significant variations in distributions of outputs between specific models, and model sub versions, with GPT-4o-Mini-2024-07-18 showing notably better performance (32-43% yes responses) compared to GPT-4-0125-preview's extreme bias (98-99% yes responses). We show that sampling methods and model sub-versions significantly impact results: repeated independent API calls produce different distributions compared to batch sampling within a single call. While no current GPT model can simultaneously achieve a uniform distribution and Markovian properties in one-shot testing, few-shot sampling can approach uniform distributions under certain conditions. We explore the Temperature parameter, providing a definition and comparative results. We further compare our results to true random binary series and test specifically for the common human bias of Negative Recency - finding LLMs have a mixed ability to 'beat' humans in this one regard. These findings emphasise the critical importance of careful LLM integration into ABMs for financial markets and more broadly.

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance significantly.

Diffusion or flow-based models are powerful generative paradigms that are notoriously hard to sample as samples are defined as solutions to high-dimensional Ordinary or Stochastic Differential Equations (ODEs/SDEs) which require a large Number of Function Evaluations (NFE) to approximate well. Existing methods to alleviate the costly sampling process include model distillation and designing dedicated ODE solvers. However, distillation is costly to train and sometimes can deteriorate quality, while dedicated solvers still require relatively large NFE to produce high quality samples. In this paper we introduce "Bespoke solvers", a novel framework for constructing custom ODE solvers tailored to the ODE of a given pre-trained flow model. Our approach optimizes an order consistent and parameter-efficient solver (e.g., with 80 learnable parameters), is trained for roughly 1% of the GPU time required for training the pre-trained model, and significantly improves approximation and generation quality compared to dedicated solvers. For example, a Bespoke solver for a CIFAR10 model produces samples with Fr\'echet Inception Distance (FID) of 2.73 with 10 NFE, and gets to 1% of the Ground Truth (GT) FID (2.59) for this model with only 20 NFE. On the more challenging ImageNet-64times64, Bespoke samples at 2.2 FID with 10 NFE, and gets within 2% of GT FID (1.71) with 20 NFE.

Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make predictions and provide uncertainty quantification for chaotic dynamical systems. In these applications, diffusion models can implicitly represent knowledge about outliers and extreme events; however, querying that knowledge through conditional sampling or measuring probabilities is surprisingly difficult. Existing methods for conditional sampling at inference time seek mainly to enforce the constraints, which is insufficient to match the statistics of the distribution or compute the probability of the chosen events. To achieve these ends, optimally one would use the conditional score function, but its computation is typically intractable. In this work, we develop a probabilistic approximation scheme for the conditional score function which provably converges to the true distribution as the noise level decreases. With this scheme we are able to sample conditionally on nonlinear userdefined events at inference time, and matches data statistics even when sampling from the tails of the distribution.

Diffusion models have demonstrated exceptional generative capabilities but are computationally intensive, posing significant challenges for deployment in resource-constrained or latency-sensitive environments. Quantization offers an effective means to reduce model size and computational cost, with post-training quantization (PTQ) being particularly appealing due to its compatibility with pre-trained models without requiring retraining or training data. However, existing PTQ methods for diffusion models often rely on architecture-specific heuristics that limit their generalizability and hinder integration with industrial deployment pipelines. To address these limitations, we propose SegQuant, a unified quantization framework that adaptively combines complementary techniques to enhance cross-model versatility. SegQuant consists of a segment-aware, graph-based quantization strategy (SegLinear) that captures structural semantics and spatial heterogeneity, along with a dual-scale quantization scheme (DualScale) that preserves polarity-asymmetric activations, which is crucial for maintaining visual fidelity in generated outputs. SegQuant is broadly applicable beyond Transformer-based diffusion models, achieving strong performance while ensuring seamless compatibility with mainstream deployment tools.

A central problem in machine learning involves modeling complex data-sets using highly flexible families of probability distributions in which learning, sampling, inference, and evaluation are still analytically or computationally tractable. Here, we develop an approach that simultaneously achieves both flexibility and tractability. The essential idea, inspired by non-equilibrium statistical physics, is to systematically and slowly destroy structure in a data distribution through an iterative forward diffusion process. We then learn a reverse diffusion process that restores structure in data, yielding a highly flexible and tractable generative model of the data. This approach allows us to rapidly learn, sample from, and evaluate probabilities in deep generative models with thousands of layers or time steps, as well as to compute conditional and posterior probabilities under the learned model. We additionally release an open source reference implementation of the algorithm.

Finite symmetric groups S_n are essential in fields such as combinatorics, physics, and chemistry. However, learning a probability distribution over S_n poses significant challenges due to its intractable size and discrete nature. In this paper, we introduce SymmetricDiffusers, a novel discrete diffusion model that simplifies the task of learning a complicated distribution over S_n by decomposing it into learning simpler transitions of the reverse diffusion using deep neural networks. We identify the riffle shuffle as an effective forward transition and provide empirical guidelines for selecting the diffusion length based on the theory of random walks on finite groups. Additionally, we propose a generalized Plackett-Luce (PL) distribution for the reverse transition, which is provably more expressive than the PL distribution. We further introduce a theoretically grounded "denoising schedule" to improve sampling and learning efficiency. Extensive experiments show that our model achieves state-of-the-art or comparable performances on solving tasks including sorting 4-digit MNIST images, jigsaw puzzles, and traveling salesman problems. Our code is released at https://github.com/DSL-Lab/SymmetricDiffusers.

Discrete diffusion or flow models could enable faster and more controllable sequence generation than autoregressive models. We show that na\"ive linear flow matching on the simplex is insufficient toward this goal since it suffers from discontinuities in the training target and further pathologies. To overcome this, we develop Dirichlet flow matching on the simplex based on mixtures of Dirichlet distributions as probability paths. In this framework, we derive a connection between the mixtures' scores and the flow's vector field that allows for classifier and classifier-free guidance. Further, we provide distilled Dirichlet flow matching, which enables one-step sequence generation with minimal performance hits, resulting in O(L) speedups compared to autoregressive models. On complex DNA sequence generation tasks, we demonstrate superior performance compared to all baselines in distributional metrics and in achieving desired design targets for generated sequences. Finally, we show that our classifier-free guidance approach improves unconditional generation and is effective for generating DNA that satisfies design targets. Code is available at https://github.com/HannesStark/dirichlet-flow-matching.

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.

Temporal Point Processes (TPP) are probabilistic generative frameworks. They model discrete event sequences localized in continuous time. Generally, real-life events reveal descriptive information, known as marks. Marked TPPs model time and marks of the event together for practical relevance. Conditioned on past events, marked TPPs aim to learn the joint distribution of the time and the mark of the next event. For simplicity, conditionally independent TPP models assume time and marks are independent given event history. They factorize the conditional joint distribution of time and mark into the product of individual conditional distributions. This structural limitation in the design of TPP models hurt the predictive performance on entangled time and mark interactions. In this work, we model the conditional inter-dependence of time and mark to overcome the limitations of conditionally independent models. We construct a multivariate TPP conditioning the time distribution on the current event mark in addition to past events. Besides the conventional intensity-based models for conditional joint distribution, we also draw on flexible intensity-free TPP models from the literature. The proposed TPP models outperform conditionally independent and dependent models in standard prediction tasks. Our experimentation on various datasets with multiple evaluation metrics highlights the merit of the proposed approach.

This paper studies distributional model risk in marginal problems, where each marginal measure is assumed to lie in a Wasserstein ball centered at a fixed reference measure with a given radius. Theoretically, we establish several fundamental results including strong duality, finiteness of the proposed Wasserstein distributional model risk, and the existence of an optimizer at each radius. In addition, we show continuity of the Wasserstein distributional model risk as a function of the radius. Using strong duality, we extend the well-known Makarov bounds for the distribution function of the sum of two random variables with given marginals to Wasserstein distributionally robust Markarov bounds. Practically, we illustrate our results on four distinct applications when the sample information comes from multiple data sources and only some marginal reference measures are identified. They are: partial identification of treatment effects; externally valid treatment choice via robust welfare functions; Wasserstein distributionally robust estimation under data combination; and evaluation of the worst aggregate risk measures.

Flow matching is a powerful framework for generating high-quality samples in various applications, especially image synthesis. However, the intensive computational demands of these models, especially during the fine-tuning process and sampling processes, pose significant challenges for low-resource scenarios. This paper introduces Bellman Optimal Step-size Straightening (BOSS) technique for distilling flow-matching generative models: it aims specifically for a few-step efficient image sampling while adhering to a computational budget constraint. First, this technique involves a dynamic programming algorithm that optimizes the step sizes of the pretrained network. Then, it refines the velocity network to match the optimal step sizes, aiming to straighten the generation paths. Extensive experimental evaluations across image generation tasks demonstrate the efficacy of BOSS in terms of both resource utilization and image quality. Our results reveal that BOSS achieves substantial gains in efficiency while maintaining competitive sample quality, effectively bridging the gap between low-resource constraints and the demanding requirements of flow-matching generative models. Our paper also fortifies the responsible development of artificial intelligence, offering a more sustainable generative model that reduces computational costs and environmental footprints. Our code can be found at https://github.com/nguyenngocbaocmt02/BOSS.

We study the problem of estimating the distribution of the return of a policy using an offline dataset that is not generated from the policy, i.e., distributional offline policy evaluation (OPE). We propose an algorithm called Fitted Likelihood Estimation (FLE), which conducts a sequence of Maximum Likelihood Estimation (MLE) and has the flexibility of integrating any state-of-the-art probabilistic generative models as long as it can be trained via MLE. FLE can be used for both finite-horizon and infinite-horizon discounted settings where rewards can be multi-dimensional vectors. Our theoretical results show that for both finite-horizon and infinite-horizon discounted settings, FLE can learn distributions that are close to the ground truth under total variation distance and Wasserstein distance, respectively. Our theoretical results hold under the conditions that the offline data covers the test policy's traces and that the supervised learning MLE procedures succeed. Experimentally, we demonstrate the performance of FLE with two generative models, Gaussian mixture models and diffusion models. For the multi-dimensional reward setting, FLE with diffusion models is capable of estimating the complicated distribution of the return of a test policy.

Safety-critical applications such as autonomous vehicles and social robots require fast computation and accurate probability density estimation on trajectory prediction. To address both requirements, this paper presents a new normalizing flow-based trajectory prediction model named FlowChain. FlowChain is a stack of conditional continuously-indexed flows (CIFs) that are expressive and allow analytical probability density computation. This analytical computation is faster than the generative models that need additional approximations such as kernel density estimation. Moreover, FlowChain is more accurate than the Gaussian mixture-based models due to fewer assumptions on the estimated density. FlowChain also allows a rapid update of estimated probability densities. This update is achieved by adopting the newest observed position and reusing the flow transformations and its log-det-jacobians that represent the motion trend. This update is completed in less than one millisecond because this reuse greatly omits the computational cost. Experimental results showed our FlowChain achieved state-of-the-art trajectory prediction accuracy compared to previous methods. Furthermore, our FlowChain demonstrated superiority in the accuracy and speed of density estimation. Our code is available at https://github.com/meaten/FlowChain-ICCV2023

Robust routing under uncertainty is central to real-world logistics, yet most benchmarks assume static, idealized settings. We present SVRPBench, the first open benchmark to capture high-fidelity stochastic dynamics in vehicle routing at urban scale. Spanning more than 500 instances with up to 1000 customers, it simulates realistic delivery conditions: time-dependent congestion, log-normal delays, probabilistic accidents, and empirically grounded time windows for residential and commercial clients. Our pipeline generates diverse, constraint-rich scenarios, including multi-depot and multi-vehicle setups. Benchmarking reveals that state-of-the-art RL solvers like POMO and AM degrade by over 20% under distributional shift, while classical and metaheuristic methods remain robust. To enable reproducible research, we release the dataset and evaluation suite. SVRPBench challenges the community to design solvers that generalize beyond synthetic assumptions and adapt to real-world uncertainty.

Wasserstein gradient flow has emerged as a promising approach to solve optimization problems over the space of probability distributions. A recent trend is to use the well-known JKO scheme in combination with input convex neural networks to numerically implement the proximal step. The most challenging step, in this setup, is to evaluate functions involving density explicitly, such as entropy, in terms of samples. This paper builds on the recent works with a slight but crucial difference: we propose to utilize a variational formulation of the objective function formulated as maximization over a parametric class of functions. Theoretically, the proposed variational formulation allows the construction of gradient flows directly for empirical distributions with a well-defined and meaningful objective function. Computationally, this approach replaces the computationally expensive step in existing methods, to handle objective functions involving density, with inner loop updates that only require a small batch of samples and scale well with the dimension. The performance and scalability of the proposed method are illustrated with the aid of several numerical experiments involving high-dimensional synthetic and real datasets.

Normalizing flows are powerful non-parametric statistical models that function as a hybrid between density estimators and generative models. Current learning algorithms for normalizing flows assume that data points are sampled independently, an assumption that is frequently violated in practice, which may lead to erroneous density estimation and data generation. We propose a likelihood objective of normalizing flows incorporating dependencies between the data points, for which we derive a flexible and efficient learning algorithm suitable for different dependency structures. We show that respecting dependencies between observations can improve empirical results on both synthetic and real-world data, and leads to higher statistical power in a downstream application to genome-wide association studies.

We consider multi-draft speculative sampling, where the proposal sequences are sampled independently from different draft models. At each step, a token-level draft selection scheme takes a list of valid tokens as input and produces an output token whose distribution matches that of the target model. Previous works have demonstrated that the optimal scheme (which maximizes the probability of accepting one of the input tokens) can be cast as a solution to a linear program. In this work we show that the optimal scheme can be decomposed into a two-step solution: in the first step an importance sampling (IS) type scheme is used to select one intermediate token; in the second step (single-draft) speculative sampling is applied to generate the output token. For the case of two identical draft models we further 1) establish a necessary and sufficient condition on the distributions of the target and draft models for the acceptance probability to equal one and 2) provide an explicit expression for the optimal acceptance probability. Our theoretical analysis also motives a new class of token-level selection scheme based on weighted importance sampling. Our experimental results demonstrate consistent improvements in the achievable block efficiency and token rates over baseline schemes in a number of scenarios.

Privacy restrictions hinder the sharing of real-world Water Distribution Network (WDN) models, limiting the application of emerging data-driven machine learning, which typically requires extensive observations. To address this challenge, we propose the dataset DiTEC-WDN that comprises 36,000 unique scenarios simulated over either short-term (24 hours) or long-term (1 year) periods. We constructed this dataset using an automated pipeline that optimizes crucial parameters (e.g., pressure, flow rate, and demand patterns), facilitates large-scale simulations, and records discrete, synthetic but hydraulically realistic states under standard conditions via rule validation and post-hoc analysis. With a total of 228 million generated graph-based states, DiTEC-WDN can support a variety of machine-learning tasks, including graph-level, node-level, and link-level regression, as well as time-series forecasting. This contribution, released under a public license, encourages open scientific research in the critical water sector, eliminates the risk of exposing sensitive data, and fulfills the need for a large-scale water distribution network benchmark for study comparisons and scenario analysis.

Machine learning models in modern mass-market applications are often updated over time. One of the foremost challenges faced is that, despite increasing overall performance, these updates may flip specific model predictions in unpredictable ways. In practice, researchers quantify the number of unstable predictions between models pre and post update -- i.e., predictive churn. In this paper, we study this effect through the lens of predictive multiplicity -- i.e., the prevalence of conflicting predictions over the set of near-optimal models (the Rashomon set). We show how traditional measures of predictive multiplicity can be used to examine expected churn over this set of prospective models -- i.e., the set of models that may be used to replace a baseline model in deployment. We present theoretical results on the expected churn between models within the Rashomon set from different perspectives. And we characterize expected churn over model updates via the Rashomon set, pairing our analysis with empirical results on real-world datasets -- showing how our approach can be used to better anticipate, reduce, and avoid churn in consumer-facing applications. Further, we show that our approach is useful even for models enhanced with uncertainty awareness.

This paper proposes a novel admission and routing scheme which takes into account arbitrarily assigned priorities for network flows. The presented approach leverages the centralized Software Defined Networking (SDN) capabilities in order to do so. Exact and heuristic approaches to the stated Priority Flow Admission and Routing (PFAR) problem are provided. The exact approach which provides an optimal solution is based on Integer Linear Programming (ILP). Given the potentially long running time required to find an exact and optimal solution, a heuristic approach is proposed; this approach is based on Genetic Algorithms (GAs). In order to effectively estimate the performance of the proposed approaches, a simulator that is capable of generating semi-random network topologies and flows has been developed. Experimental results for large problem instances (up 50 network nodes and thousands of network flows), show that: i) an optimal solution can be often found in few seconds (even milliseconds), and ii) the heuristic approach yields close-to-optimal solutions (approximately 95\% of the optimal) in a fixed amount of time; these experimental results demonstrate the pertinence of the proposed approaches.

Diffusion models have demonstrated remarkable generation quality but at the cost of numerous function evaluations. Recently, advanced ODE-based solvers have been developed to mitigate the substantial computational demands of reverse-diffusion solving under limited sampling steps. However, these solvers, heavily inspired by Adams-like multistep methods, rely solely on t-related Lagrange interpolation. We show that t-related Lagrange interpolation is suboptimal for diffusion model and reveal a compact search space comprised of time steps and solver coefficients. Building on our analysis, we propose a novel differentiable solver search algorithm to identify more optimal solver. Equipped with the searched solver, rectified-flow models, e.g., SiT-XL/2 and FlowDCN-XL/2, achieve FID scores of 2.40 and 2.35, respectively, on ImageNet256 with only 10 steps. Meanwhile, DDPM model, DiT-XL/2, reaches a FID score of 2.33 with only 10 steps. Notably, our searched solver outperforms traditional solvers by a significant margin. Moreover, our searched solver demonstrates generality across various model architectures, resolutions, and model sizes.

Discrete diffusion has recently emerged as a promising paradigm in discrete data modeling. However, existing methods typically rely on a fixed rate transition matrix during training, which not only limits the expressiveness of latent representations, a fundamental strength of variational methods, but also constrains the overall design space. To address these limitations, we propose Discrete Markov Bridge, a novel framework specifically designed for discrete representation learning. Our approach is built upon two key components: Matrix Learning and Score Learning. We conduct a rigorous theoretical analysis, establishing formal performance guarantees for Matrix Learning and proving the convergence of the overall framework. Furthermore, we analyze the space complexity of our method, addressing practical constraints identified in prior studies. Extensive empirical evaluations validate the effectiveness of the proposed Discrete Markov Bridge, which achieves an Evidence Lower Bound (ELBO) of 1.38 on the Text8 dataset, outperforming established baselines. Moreover, the proposed model demonstrates competitive performance on the CIFAR-10 dataset, achieving results comparable to those obtained by image-specific generation approaches.

Masked diffusion models (MDMs) have emerged as a popular research topic for generative modeling of discrete data, thanks to their superior performance over other discrete diffusion models, and are rivaling the auto-regressive models (ARMs) for language modeling tasks. The recent effort in simplifying the masked diffusion framework further leads to alignment with continuous-space diffusion models and more principled training and sampling recipes. In this paper, however, we reveal that both training and sampling of MDMs are theoretically free from the time variable, arguably the key signature of diffusion models, and are instead equivalent to masked models. The connection on the sampling aspect is drawn by our proposed first-hitting sampler (FHS). Specifically, we show that the FHS is theoretically equivalent to MDMs' original generation process while significantly alleviating the time-consuming categorical sampling and achieving a 20times speedup. In addition, our investigation raises doubts about whether MDMs can truly beat ARMs. We identify, for the first time, an underlying numerical issue, even with the commonly used 32-bit floating-point precision, which results in inaccurate categorical sampling. We show that the numerical issue lowers the effective temperature both theoretically and empirically, and the resulting decrease in token diversity makes previous evaluations, which assess the generation quality solely through the incomplete generative perplexity metric, somewhat unfair.

The performance of a language model has been shown to be effectively modeled as a power-law in its parameter count. Here we study the scaling behaviors of Routing Networks: architectures that conditionally use only a subset of their parameters while processing an input. For these models, parameter count and computational requirement form two independent axes along which an increase leads to better performance. In this work we derive and justify scaling laws defined on these two variables which generalize those known for standard language models and describe the performance of a wide range of routing architectures trained via three different techniques. Afterwards we provide two applications of these laws: first deriving an Effective Parameter Count along which all models scale at the same rate, and then using the scaling coefficients to give a quantitative comparison of the three routing techniques considered. Our analysis derives from an extensive evaluation of Routing Networks across five orders of magnitude of size, including models with hundreds of experts and hundreds of billions of parameters.

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

Score-based generative modeling with probability flow ordinary differential equations (ODEs) has achieved remarkable success in a variety of applications. While various fast ODE-based samplers have been proposed in the literature and employed in practice, the theoretical understandings about convergence properties of the probability flow ODE are still quite limited. In this paper, we provide the first non-asymptotic convergence analysis for a general class of probability flow ODE samplers in 2-Wasserstein distance, assuming accurate score estimates. We then consider various examples and establish results on the iteration complexity of the corresponding ODE-based samplers.

Generative flow networks (GFlowNets) are a family of algorithms that learn a generative policy to sample discrete objects x with non-negative reward R(x). Learning objectives guarantee the GFlowNet samples x from the target distribution p^*(x) propto R(x) when loss is globally minimized over all states or trajectories, but it is unclear how well they perform with practical limits on training resources. We introduce an efficient evaluation strategy to compare the learned sampling distribution to the target reward distribution. As flows can be underdetermined given training data, we clarify the importance of learned flows to generalization and matching p^*(x) in practice. We investigate how to learn better flows, and propose (i) prioritized replay training of high-reward x, (ii) relative edge flow policy parametrization, and (iii) a novel guided trajectory balance objective, and show how it can solve a substructure credit assignment problem. We substantially improve sample efficiency on biochemical design tasks.

High-precision modeling of systems is one of the main areas of industrial data analysis. Models of systems, their digital twins, are used to predict their behavior under various conditions. We have developed several models of a storage system using machine learning-based generative models. The system consists of several components: hard disk drive (HDD) and solid-state drive (SSD) storage pools with different RAID schemes and cache. Each storage component is represented by a probabilistic model that describes the probability distribution of the component performance in terms of IOPS and latency, depending on their configuration and external data load parameters. The results of the experiments demonstrate the errors of 4-10 % for IOPS and 3-16 % for latency predictions depending on the components and models of the system. The predictions show up to 0.99 Pearson correlation with Little's law, which can be used for unsupervised reliability checks of the models. In addition, we present novel data sets that can be used for benchmarking regression algorithms, conditional generative models, and uncertainty estimation methods in machine learning.

Diffusion models have exhibited excellent performance in various domains. The probability flow ordinary differential equation (ODE) of diffusion models (i.e., diffusion ODEs) is a particular case of continuous normalizing flows (CNFs), which enables deterministic inference and exact likelihood evaluation. However, the likelihood estimation results by diffusion ODEs are still far from those of the state-of-the-art likelihood-based generative models. In this work, we propose several improved techniques for maximum likelihood estimation for diffusion ODEs, including both training and evaluation perspectives. For training, we propose velocity parameterization and explore variance reduction techniques for faster convergence. We also derive an error-bounded high-order flow matching objective for finetuning, which improves the ODE likelihood and smooths its trajectory. For evaluation, we propose a novel training-free truncated-normal dequantization to fill the training-evaluation gap commonly existing in diffusion ODEs. Building upon these techniques, we achieve state-of-the-art likelihood estimation results on image datasets (2.56 on CIFAR-10, 3.43/3.69 on ImageNet-32) without variational dequantization or data augmentation.

The development of video diffusion models unveils a significant challenge: the substantial computational demands. To mitigate this challenge, we note that the reverse process of diffusion exhibits an inherent entropy-reducing nature. Given the inter-frame redundancy in video modality, maintaining full frame rates in high-entropy stages is unnecessary. Based on this insight, we propose TPDiff, a unified framework to enhance training and inference efficiency. By dividing diffusion into several stages, our framework progressively increases frame rate along the diffusion process with only the last stage operating on full frame rate, thereby optimizing computational efficiency. To train the multi-stage diffusion model, we introduce a dedicated training framework: stage-wise diffusion. By solving the partitioned probability flow ordinary differential equations (ODE) of diffusion under aligned data and noise, our training strategy is applicable to various diffusion forms and further enhances training efficiency. Comprehensive experimental evaluations validate the generality of our method, demonstrating 50% reduction in training cost and 1.5x improvement in inference efficiency.

Diffusion models have emerged as dominant performers for image generation. To support training large diffusion models, this paper studies pipeline parallel training of diffusion models and proposes DiffusionPipe, a synchronous pipeline training system that advocates innovative pipeline bubble filling technique, catering to structural characteristics of diffusion models. State-of-the-art diffusion models typically include trainable (the backbone) and non-trainable (e.g., frozen input encoders) parts. We first unify optimal stage partitioning and pipeline scheduling of single and multiple backbones in representative diffusion models with a dynamic programming approach. We then propose to fill the computation of non-trainable model parts into idle periods of the pipeline training of the backbones by an efficient greedy algorithm, thus achieving high training throughput. Extensive experiments show that DiffusionPipe can achieve up to 1.41x speedup over pipeline parallel methods and 1.28x speedup over data parallel training on popular diffusion models.

Applying diffusion models in reinforcement learning for long-term planning has gained much attention recently. Several diffusion-based methods have successfully leveraged the modeling capabilities of diffusion for arbitrary distributions. These methods generate subsequent trajectories for planning and have demonstrated significant improvement. However, these methods are limited by their plain base distributions and their overlooking of the diversity of samples, in which different states have different returns. They simply leverage diffusion to learn the distribution of offline dataset, generate the trajectories whose states share the same distribution with the offline dataset. As a result, the probability of these models reaching the high-return states is largely dependent on the dataset distribution. Even equipped with the guidance model, the performance is still suppressed. To address these limitations, in this paper, we propose a novel method called CDiffuser, which devises a return contrast mechanism to pull the states in generated trajectories towards high-return states while pushing them away from low-return states to improve the base distribution. Experiments on 14 commonly used D4RL benchmarks demonstrate the effectiveness of our proposed method.

While diffusion models can successfully generate data and make predictions, they are predominantly designed for static images. We propose an approach for efficiently training diffusion models for probabilistic spatiotemporal forecasting, where generating stable and accurate rollout forecasts remains challenging, Our method, DYffusion, leverages the temporal dynamics in the data, directly coupling it with the diffusion steps in the model. We train a stochastic, time-conditioned interpolator and a forecaster network that mimic the forward and reverse processes of standard diffusion models, respectively. DYffusion naturally facilitates multi-step and long-range forecasting, allowing for highly flexible, continuous-time sampling trajectories and the ability to trade-off performance with accelerated sampling at inference time. In addition, the dynamics-informed diffusion process in DYffusion imposes a strong inductive bias and significantly improves computational efficiency compared to traditional Gaussian noise-based diffusion models. Our approach performs competitively on probabilistic forecasting of complex dynamics in sea surface temperatures, Navier-Stokes flows, and spring mesh systems.

We present MotionDiffuser, a diffusion based representation for the joint distribution of future trajectories over multiple agents. Such representation has several key advantages: first, our model learns a highly multimodal distribution that captures diverse future outcomes. Second, the simple predictor design requires only a single L2 loss training objective, and does not depend on trajectory anchors. Third, our model is capable of learning the joint distribution for the motion of multiple agents in a permutation-invariant manner. Furthermore, we utilize a compressed trajectory representation via PCA, which improves model performance and allows for efficient computation of the exact sample log probability. Subsequently, we propose a general constrained sampling framework that enables controlled trajectory sampling based on differentiable cost functions. This strategy enables a host of applications such as enforcing rules and physical priors, or creating tailored simulation scenarios. MotionDiffuser can be combined with existing backbone architectures to achieve top motion forecasting results. We obtain state-of-the-art results for multi-agent motion prediction on the Waymo Open Motion Dataset.

Diffusion models have greatly improved visual generation but are hindered by slow generation speed due to the computationally intensive nature of solving generative ODEs. Rectified flow, a widely recognized solution, improves generation speed by straightening the ODE path. Its key components include: 1) using the diffusion form of flow-matching, 2) employing boldsymbol v-prediction, and 3) performing rectification (a.k.a. reflow). In this paper, we argue that the success of rectification primarily lies in using a pretrained diffusion model to obtain matched pairs of noise and samples, followed by retraining with these matched noise-sample pairs. Based on this, components 1) and 2) are unnecessary. Furthermore, we highlight that straightness is not an essential training target for rectification; rather, it is a specific case of flow-matching models. The more critical training target is to achieve a first-order approximate ODE path, which is inherently curved for models like DDPM and Sub-VP. Building on this insight, we propose Rectified Diffusion, which generalizes the design space and application scope of rectification to encompass the broader category of diffusion models, rather than being restricted to flow-matching models. We validate our method on Stable Diffusion v1-5 and Stable Diffusion XL. Our method not only greatly simplifies the training procedure of rectified flow-based previous works (e.g., InstaFlow) but also achieves superior performance with even lower training cost. Our code is available at https://github.com/G-U-N/Rectified-Diffusion.

Out-of-distribution (OOD) detection is a critical requirement for the deployment of deep neural networks. This paper introduces the HEAT model, a new post-hoc OOD detection method estimating the density of in-distribution (ID) samples using hybrid energy-based models (EBM) in the feature space of a pre-trained backbone. HEAT complements prior density estimators of the ID density, e.g. parametric models like the Gaussian Mixture Model (GMM), to provide an accurate yet robust density estimation. A second contribution is to leverage the EBM framework to provide a unified density estimation and to compose several energy terms. Extensive experiments demonstrate the significance of the two contributions. HEAT sets new state-of-the-art OOD detection results on the CIFAR-10 / CIFAR-100 benchmark as well as on the large-scale Imagenet benchmark. The code is available at: https://github.com/MarcLafon/heatood.

Recent progress in neural forecasting accelerated improvements in the performance of large-scale forecasting systems. Yet, long-horizon forecasting remains a very difficult task. Two common challenges afflicting the task are the volatility of the predictions and their computational complexity. We introduce N-HiTS, a model which addresses both challenges by incorporating novel hierarchical interpolation and multi-rate data sampling techniques. These techniques enable the proposed method to assemble its predictions sequentially, emphasizing components with different frequencies and scales while decomposing the input signal and synthesizing the forecast. We prove that the hierarchical interpolation technique can efficiently approximate arbitrarily long horizons in the presence of smoothness. Additionally, we conduct extensive large-scale dataset experiments from the long-horizon forecasting literature, demonstrating the advantages of our method over the state-of-the-art methods, where N-HiTS provides an average accuracy improvement of almost 20% over the latest Transformer architectures while reducing the computation time by an order of magnitude (50 times). Our code is available at bit.ly/3VA5DoT

Diffusion models (DMs) represent state-of-the-art generative models for continuous inputs. DMs work by constructing a Stochastic Differential Equation (SDE) in the input space (ie, position space), and using a neural network to reverse it. In this work, we introduce a novel generative modeling framework grounded in phase space dynamics, where a phase space is defined as {an augmented space encompassing both position and velocity.} Leveraging insights from Stochastic Optimal Control, we construct a path measure in the phase space that enables efficient sampling. {In contrast to DMs, our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.} This early prediction sets the stage for efficient data generation by leveraging additional velocity information along the trajectory. On standard image generation benchmarks, our model yields favorable performance over baselines in the regime of small Number of Function Evaluations (NFEs). Furthermore, our approach rivals the performance of diffusion models equipped with efficient sampling techniques, underscoring its potential as a new tool generative modeling.

Straightening the probability flow of the continuous-time generative models, such as diffusion models or flow-based models, is the key to fast sampling through the numerical solvers, existing methods learn a linear path by directly generating the probability path the joint distribution between the noise and data distribution. One key reason for the slow sampling speed of the ODE-based solvers that simulate these generative models is the global truncation error of the ODE solver, caused by the high curvature of the ODE trajectory, which explodes the truncation error of the numerical solvers in the low-NFE regime. To address this challenge, We propose a novel method called SeqRF, a learning technique that straightens the probability flow to reduce the global truncation error and hence enable acceleration of sampling and improve the synthesis quality. In both theoretical and empirical studies, we first observe the straightening property of our SeqRF. Through empirical evaluations via SeqRF over flow-based generative models, We achieve surpassing results on CIFAR-10, CelebA-64 times 64, and LSUN-Church datasets.

We present SwD, a scale-wise distillation framework for diffusion models (DMs), which effectively employs next-scale prediction ideas for diffusion-based few-step generators. In more detail, SwD is inspired by the recent insights relating diffusion processes to the implicit spectral autoregression. We suppose that DMs can initiate generation at lower data resolutions and gradually upscale the samples at each denoising step without loss in performance while significantly reducing computational costs. SwD naturally integrates this idea into existing diffusion distillation methods based on distribution matching. Also, we enrich the family of distribution matching approaches by introducing a novel patch loss enforcing finer-grained similarity to the target distribution. When applied to state-of-the-art text-to-image diffusion models, SwD approaches the inference times of two full resolution steps and significantly outperforms the counterparts under the same computation budget, as evidenced by automated metrics and human preference studies.

In this paper, we establish a connection between the parameterization of flow-based and energy-based generative models, and present a new flow-based modeling approach called energy-based normalizing flow (EBFlow). We demonstrate that by optimizing EBFlow with score-matching objectives, the computation of Jacobian determinants for linear transformations can be entirely bypassed. This feature enables the use of arbitrary linear layers in the construction of flow-based models without increasing the computational time complexity of each training iteration from O(D^2L) to O(D^3L) for an L-layered model that accepts D-dimensional inputs. This makes the training of EBFlow more efficient than the commonly-adopted maximum likelihood training method. In addition to the reduction in runtime, we enhance the training stability and empirical performance of EBFlow through a number of techniques developed based on our analysis of the score-matching methods. The experimental results demonstrate that our approach achieves a significant speedup compared to maximum likelihood estimation while outperforming prior methods with a noticeable margin in terms of negative log-likelihood (NLL).

Combining several (sample approximations of) distributions, which we term sub-posteriors, into a single distribution proportional to their product, is a common challenge. Occurring, for instance, in distributed 'big data' problems, or when working under multi-party privacy constraints. Many existing approaches resort to approximating the individual sub-posteriors for practical necessity, then find either an analytical approximation or sample approximation of the resulting (product-pooled) posterior. The quality of the posterior approximation for these approaches is poor when the sub-posteriors fall out-with a narrow range of distributional form, such as being approximately Gaussian. Recently, a Fusion approach has been proposed which finds an exact Monte Carlo approximation of the posterior, circumventing the drawbacks of approximate approaches. Unfortunately, existing Fusion approaches have a number of computational limitations, particularly when unifying a large number of sub-posteriors. In this paper, we generalise the theory underpinning existing Fusion approaches, and embed the resulting methodology within a recursive divide-and-conquer sequential Monte Carlo paradigm. This ultimately leads to a competitive Fusion approach, which is robust to increasing numbers of sub-posteriors.

Predicting the intermediate trajectories between an initial and target distribution is a central problem in generative modeling. Existing approaches, such as flow matching and Schr\"odinger Bridge Matching, effectively learn mappings between two distributions by modeling a single stochastic path. However, these methods are inherently limited to unimodal transitions and cannot capture branched or divergent evolution from a common origin to multiple distinct outcomes. To address this, we introduce Branched Schr\"odinger Bridge Matching (BranchSBM), a novel framework that learns branched Schr\"odinger bridges. BranchSBM parameterizes multiple time-dependent velocity fields and growth processes, enabling the representation of population-level divergence into multiple terminal distributions. We show that BranchSBM is not only more expressive but also essential for tasks involving multi-path surface navigation, modeling cell fate bifurcations from homogeneous progenitor states, and simulating diverging cellular responses to perturbations.

Diffusion models and Flow Matching generate high-quality samples but are slow at inference, and distilling them into few-step models often leads to instability and extensive tuning. To resolve these trade-offs, we propose Inductive Moment Matching (IMM), a new class of generative models for one- or few-step sampling with a single-stage training procedure. Unlike distillation, IMM does not require pre-training initialization and optimization of two networks; and unlike Consistency Models, IMM guarantees distribution-level convergence and remains stable under various hyperparameters and standard model architectures. IMM surpasses diffusion models on ImageNet-256x256 with 1.99 FID using only 8 inference steps and achieves state-of-the-art 2-step FID of 1.98 on CIFAR-10 for a model trained from scratch.

This paper presents a new exploration into a category of diffusion models built upon state space architecture. We endeavor to train diffusion models for image data, wherein the traditional U-Net backbone is supplanted by a state space backbone, functioning on raw patches or latent space. Given its notable efficacy in accommodating long-range dependencies, Diffusion State Space Models (DiS) are distinguished by treating all inputs including time, condition, and noisy image patches as tokens. Our assessment of DiS encompasses both unconditional and class-conditional image generation scenarios, revealing that DiS exhibits comparable, if not superior, performance to CNN-based or Transformer-based U-Net architectures of commensurate size. Furthermore, we analyze the scalability of DiS, gauged by the forward pass complexity quantified in Gflops. DiS models with higher Gflops, achieved through augmentation of depth/width or augmentation of input tokens, consistently demonstrate lower FID. In addition to demonstrating commendable scalability characteristics, DiS-H/2 models in latent space achieve performance levels akin to prior diffusion models on class-conditional ImageNet benchmarks at the resolution of 256times256 and 512times512, while significantly reducing the computational burden. The code and models are available at: https://github.com/feizc/DiS.

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.

In this paper, we introduce a new class of score-based generative models (SGMs) designed to handle high-cardinality data distributions by leveraging concepts from mean-field theory. We present mean-field chaos diffusion models (MF-CDMs), which address the curse of dimensionality inherent in high-cardinality data by utilizing the propagation of chaos property of interacting particles. By treating high-cardinality data as a large stochastic system of interacting particles, we develop a novel score-matching method for infinite-dimensional chaotic particle systems and propose an approximation scheme that employs a subdivision strategy for efficient training. Our theoretical and empirical results demonstrate the scalability and effectiveness of MF-CDMs for managing large high-cardinality data structures, such as 3D point clouds.

This study addresses challenges in reverse logistics, a frequently overlooked but essential component of last-mile delivery, particularly in disaster relief scenarios where infrastructure disruptions demand adaptive solutions. While hub-and-spoke logistics networks excel at long-distance scalability, they often fail to optimize closely spaced spokes reliant on distant hubs, introducing inefficiencies in transit times and resource allocation. Using 20 Apple AirTags embedded in packages, this research provides empirical insights into logistical flows, capturing granular spatial and temporal data through Bluetooth LE (BLE) 5 trackers integrated with the Apple Find My network. These trackers demonstrated their value in monitoring dynamic cargo movements, enabling real-time adjustments in mobile hub placement and route optimization, particularly in disaster relief contexts like Hurricane Helene. A novel application of discrete event simulation (DES) further explored the saddle point in hub-spoke configurations, where excessive hub reliance clashes with diminishing spoke interaction demand. By coupling simulation results with empirical AirTag tracking, the study highlights the potential of BLE technology to refine reverse logistics, reduce delays, and improve operational flexibility in both routine and crisis-driven delivery networks.

Recently, diffusion probabilistic models (DPMs) have achieved promising results in diverse generative tasks. A typical DPM framework includes a forward process that gradually diffuses the data distribution and a reverse process that recovers the data distribution from time-dependent data scores. In this work, we observe that the stochastic reverse process of data scores is a martingale, from which concentration bounds and the optional stopping theorem for data scores can be derived. Then, we discover a simple way for calibrating an arbitrary pretrained DPM, with which the score matching loss can be reduced and the lower bounds of model likelihood can consequently be increased. We provide general calibration guidelines under various model parametrizations. Our calibration method is performed only once and the resulting models can be used repeatedly for sampling. We conduct experiments on multiple datasets to empirically validate our proposal. Our code is at https://github.com/thudzj/Calibrated-DPMs.

MATSim (Multi-Agent Transport Simulation Toolkit) is an open source large-scale agent-based transportation planning project applied to various areas like road transport, public transport, freight transport, regional evacuation, etc. BEAM (Behavior, Energy, Autonomy, and Mobility) framework extends MATSim to enable powerful and scalable analysis of urban transportation systems. The agents from the BEAM simulation exhibit 'mode choice' behavior based on multinomial logit model. In our study, we consider eight mode choices viz. bike, car, walk, ride hail, driving to transit, walking to transit, ride hail to transit, and ride hail pooling. The 'alternative specific constants' for each mode choice are critical hyperparameters in a configuration file related to a particular scenario under experimentation. We use the 'Urbansim-10k' BEAM scenario (with 10,000 population size) for all our experiments. Since these hyperparameters affect the simulation in complex ways, manual calibration methods are time consuming. We present a parallel Bayesian optimization method with early stopping rule to achieve fast convergence for the given multi-in-multi-out problem to its optimal configurations. Our model is based on an open source HpBandSter package. This approach combines hierarchy of several 1D Kernel Density Estimators (KDE) with a cheap evaluator (Hyperband, a single multidimensional KDE). Our model has also incorporated extrapolation based early stopping rule. With our model, we could achieve a 25% L1 norm for a large-scale BEAM simulation in fully autonomous manner. To the best of our knowledge, our work is the first of its kind applied to large-scale multi-agent transportation simulations. This work can be useful for surrogate modeling of scenarios with very large populations.

Normalizing flows can transform a simple prior probability distribution into a more complex target distribution. Here, we evaluate the ability and efficiency of generative machine learning methods to sample the Boltzmann distribution of an atomistic model for glass-forming liquids. This is a notoriously difficult task, as it amounts to ergodically exploring the complex free energy landscape of a disordered and frustrated many-body system. We optimize a normalizing flow model to successfully transform high-temperature configurations of a dense liquid into low-temperature ones, near the glass transition. We perform a detailed comparative analysis with established enhanced sampling techniques developed in the physics literature to assess and rank the performance of normalizing flows against state-of-the-art algorithms. We demonstrate that machine learning methods are very promising, showing a large speedup over conventional molecular dynamics. Normalizing flows show performances comparable to parallel tempering and population annealing, while still falling far behind the swap Monte Carlo algorithm. Our study highlights the potential of generative machine learning models in scientific computing for complex systems, but also points to some of its current limitations and the need for further improvement.

Diffusion models achieve strong generative performance but often rely on large datasets that may include sensitive content. This challenge is compounded by the models' tendency to memorize training data, raising privacy concerns. SFBD (Lu et al., 2025) addresses this by training on corrupted data and using limited clean samples to capture local structure and improve convergence. However, its iterative denoising and fine-tuning loop requires manual coordination, making it burdensome to implement. We reinterpret SFBD as an alternating projection algorithm and introduce a continuous variant, SFBD flow, that removes the need for alternating steps. We further show its connection to consistency constraint-based methods, and demonstrate that its practical instantiation, Online SFBD, consistently outperforms strong baselines across benchmarks.

Flow matching is a recent state-of-the-art framework for generative modeling based on ordinary differential equations (ODEs). While closely related to diffusion models, it provides a more general perspective on generative modeling. Although inverse problem solving has been extensively explored using diffusion models, it has not been rigorously examined within the broader context of flow models. Therefore, here we extend the diffusion inverse solvers (DIS) - which perform posterior sampling by combining a denoising diffusion prior with an likelihood gradient - into the flow framework. Specifically, by driving the flow-version of Tweedie's formula, we decompose the flow ODE into two components: one for clean image estimation and the other for noise estimation. By integrating the likelihood gradient and stochastic noise into each component, respectively, we demonstrate that posterior sampling for inverse problem solving can be effectively achieved using flows. Our proposed solver, Flow-Driven Posterior Sampling (FlowDPS), can also be seamlessly integrated into a latent flow model with a transformer architecture. Across four linear inverse problems, we confirm that FlowDPS outperforms state-of-the-art alternatives, all without requiring additional training.

We present AROMA (Attentive Reduced Order Model with Attention), a framework designed to enhance the modeling of partial differential equations (PDEs) using local neural fields. Our flexible encoder-decoder architecture can obtain smooth latent representations of spatial physical fields from a variety of data types, including irregular-grid inputs and point clouds. This versatility eliminates the need for patching and allows efficient processing of diverse geometries. The sequential nature of our latent representation can be interpreted spatially and permits the use of a conditional transformer for modeling the temporal dynamics of PDEs. By employing a diffusion-based formulation, we achieve greater stability and enable longer rollouts compared to conventional MSE training. AROMA's superior performance in simulating 1D and 2D equations underscores the efficacy of our approach in capturing complex dynamical behaviors.

Diffusion bridges have shown potential in paired image-to-image (I2I) translation tasks. However, existing methods are limited by their unidirectional nature, requiring separate models for forward and reverse translations. This not only doubles the computational cost but also restricts their practicality. In this work, we introduce the Bidirectional Diffusion Bridge Model (BDBM), a scalable approach that facilitates bidirectional translation between two coupled distributions using a single network. BDBM leverages the Chapman-Kolmogorov Equation for bridges, enabling it to model data distribution shifts across timesteps in both forward and backward directions by exploiting the interchangeability of the initial and target timesteps within this framework. Notably, when the marginal distribution given endpoints is Gaussian, BDBM's transition kernels in both directions possess analytical forms, allowing for efficient learning with a single network. We demonstrate the connection between BDBM and existing bridge methods, such as Doob's h-transform and variational approaches, and highlight its advantages. Extensive experiments on high-resolution I2I translation tasks demonstrate that BDBM not only enables bidirectional translation with minimal additional cost but also outperforms state-of-the-art bridge models. Our source code is available at [https://github.com/kvmduc/BDBM||https://github.com/kvmduc/BDBM].

Generative Flow Networks (GFlowNets) have been introduced as a method to sample a diverse set of candidates with probabilities proportional to a given reward. However, GFlowNets can only be used with a predefined scalar reward, which can be either computationally expensive or not directly accessible, in the case of multi-objective optimization (MOO) tasks for example. Moreover, to prioritize identifying high-reward candidates, the conventional practice is to raise the reward to a higher exponent, the optimal choice of which may vary across different environments. To address these issues, we propose Order-Preserving GFlowNets (OP-GFNs), which sample with probabilities in proportion to a learned reward function that is consistent with a provided (partial) order on the candidates, thus eliminating the need for an explicit formulation of the reward function. We theoretically prove that the training process of OP-GFNs gradually sparsifies the learned reward landscape in single-objective maximization tasks. The sparsification concentrates on candidates of a higher hierarchy in the ordering, ensuring exploration at the beginning and exploitation towards the end of the training. We demonstrate OP-GFN's state-of-the-art performance in single-objective maximization (totally ordered) and multi-objective Pareto front approximation (partially ordered) tasks, including synthetic datasets, molecule generation, and neural architecture search.

Dataset Distillation (DD) emerges as a powerful strategy to encapsulate the expansive information of large datasets into significantly smaller, synthetic equivalents, thereby preserving model performance with reduced computational overhead. Pursuing this objective, we introduce the Wasserstein distance, a metric grounded in optimal transport theory, to enhance distribution matching in DD. Our approach employs the Wasserstein barycenter to provide a geometrically meaningful method for quantifying distribution differences and capturing the centroid of distribution sets efficiently. By embedding synthetic data in the feature spaces of pretrained classification models, we facilitate effective distribution matching that leverages prior knowledge inherent in these models. Our method not only maintains the computational advantages of distribution matching-based techniques but also achieves new state-of-the-art performance across a range of high-resolution datasets. Extensive testing demonstrates the effectiveness and adaptability of our method, underscoring the untapped potential of Wasserstein metrics in dataset distillation.

Autoregressive language models, despite their impressive capabilities, struggle with complex reasoning and long-term planning tasks. We introduce discrete diffusion models as a novel solution to these challenges. Through the lens of subgoal imbalance, we demonstrate how diffusion models effectively learn difficult subgoals that elude autoregressive approaches. We propose Multi-granularity Diffusion Modeling (MDM), which prioritizes subgoals based on difficulty during learning. On complex tasks like Countdown, Sudoku, and Boolean Satisfiability Problems, MDM significantly outperforms autoregressive models without using search techniques. For instance, MDM achieves 91.5\% and 100\% accuracy on Countdown and Sudoku, respectively, compared to 45.8\% and 20.7\% for autoregressive models. Our work highlights the potential of diffusion-based approaches in advancing AI capabilities for sophisticated language understanding and problem-solving tasks.

We introduce Functional Diffusion Processes (FDPs), which generalize score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several extensions to derive practical training objectives. These include infinite-dimensional versions of Girsanov theorem, in order to be able to compute an ELBO, and of the sampling theorem, in order to guarantee that functional evaluations in a countable set of points are equivalent to infinite-dimensional functions. We use FDPs to build a new breed of generative models in function spaces, which do not require specialized network architectures, and that can work with any kind of continuous data. Our results on real data show that FDPs achieve high-quality image generation, using a simple MLP architecture with orders of magnitude fewer parameters than existing diffusion models.

We introduce Sundial, a family of native, flexible, and scalable time series foundation models. To predict the next-patch's distribution, we propose a TimeFlow Loss based on flow-matching, which facilitates native pre-training of Transformers on time series without discrete tokenization. Conditioned on arbitrary-length time series, our model is pre-trained without specifying any prior distribution and can generate multiple probable predictions, achieving flexibility in representation learning beyond using parametric densities. Towards time series foundation models, we leverage minimal but crucial adaptations of Transformers and curate TimeBench with 1 trillion time points, comprising mostly real-world datasets and synthetic data. By mitigating mode collapse through TimeFlow Loss, we pre-train a family of Sundial models on TimeBench, which exhibit unprecedented model capacity and generalization performance on zero-shot forecasting. In addition to presenting good scaling behavior, Sundial achieves new state-of-the-art on both point forecasting and probabilistic forecasting benchmarks. We believe that Sundial's pioneering generative paradigm will facilitate a wide variety of forecasting scenarios.

Large-scale distributed training in production datacenters constitutes a challenging workload bottlenecked by network communication. In response, both major industry players (e.g., Ultra Ethernet Consortium) and parts of academia have surprisingly, and almost unanimously, agreed that packet spraying is necessary to improve the performance of large-scale distributed training workloads. In this paper, we challenge this prevailing belief and pose the question: How close can a singlepath transport approach an optimal multipath transport? We demonstrate that singlepath transport (from a NIC's perspective) is sufficient and can perform nearly as well as an ideal multipath transport with packet spraying, particularly in the context of distributed training in leaf-spine topologies. Our assertion is based on four key observations about workloads driven by collective communication patterns: (i) flows within a collective start almost simultaneously, (ii) flow sizes are nearly equal, (iii) the completion time of a collective is more crucial than individual flow completion times, and (iv) flows can be split upon arrival. We analytically prove that singlepath transport, using minimal flow splitting (at the application layer), is equivalent to an ideal multipath transport with packet spraying in terms of maximum congestion. Our preliminary evaluations support our claims. This paper suggests an alternative agenda for developing next-generation transport protocols tailored for large-scale distributed training.

We propose a new approach for propagating stable probability distributions through neural networks. Our method is based on local linearization, which we show to be an optimal approximation in terms of total variation distance for the ReLU non-linearity. This allows propagating Gaussian and Cauchy input uncertainties through neural networks to quantify their output uncertainties. To demonstrate the utility of propagating distributions, we apply the proposed method to predicting calibrated confidence intervals and selective prediction on out-of-distribution data. The results demonstrate a broad applicability of propagating distributions and show the advantages of our method over other approaches such as moment matching.

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.

Image generation models have achieved widespread applications. As an instance, the TarFlow model combines the transformer architecture with Normalizing Flow models, achieving state-of-the-art results on multiple benchmarks. However, due to the causal form of attention requiring sequential computation, TarFlow's sampling process is extremely slow. In this paper, we demonstrate that through a series of optimization strategies, TarFlow sampling can be greatly accelerated by using the Gauss-Seidel-Jacobi (abbreviated as GS-Jacobi) iteration method. Specifically, we find that blocks in the TarFlow model have varying importance: a small number of blocks play a major role in image generation tasks, while other blocks contribute relatively little; some blocks are sensitive to initial values and prone to numerical overflow, while others are relatively robust. Based on these two characteristics, we propose the Convergence Ranking Metric (CRM) and the Initial Guessing Metric (IGM): CRM is used to identify whether a TarFlow block is "simple" (converges in few iterations) or "tough" (requires more iterations); IGM is used to evaluate whether the initial value of the iteration is good. Experiments on four TarFlow models demonstrate that GS-Jacobi sampling can significantly enhance sampling efficiency while maintaining the quality of generated images (measured by FID), achieving speed-ups of 4.53x in Img128cond, 5.32x in AFHQ, 2.96x in Img64uncond, and 2.51x in Img64cond without degrading FID scores or sample quality. Code and checkpoints are accessible on https://github.com/encoreus/GS-Jacobi_for_TarFlow

We initiate the study of statistical inference and A/B testing for first-price pacing equilibria (FPPE). The FPPE model captures the dynamics resulting from large-scale first-price auction markets where buyers use pacing-based budget management. Such markets arise in the context of internet advertising, where budgets are prevalent. We propose a statistical framework for the FPPE model, in which a limit FPPE with a continuum of items models the long-run steady-state behavior of the auction platform, and an observable FPPE consisting of a finite number of items provides the data to estimate primitives of the limit FPPE, such as revenue, Nash social welfare (a fair metric of efficiency), and other parameters of interest. We develop central limit theorems and asymptotically valid confidence intervals. Furthermore, we establish the asymptotic local minimax optimality of our estimators. We then show that the theory can be used for conducting statistically valid A/B testing on auction platforms. Numerical simulations verify our central limit theorems, and empirical coverage rates for our confidence intervals agree with our theory.

Generating photos satisfying multiple constraints find broad utility in the content creation industry. A key hurdle to accomplishing this task is the need for paired data consisting of all modalities (i.e., constraints) and their corresponding output. Moreover, existing methods need retraining using paired data across all modalities to introduce a new condition. This paper proposes a solution to this problem based on denoising diffusion probabilistic models (DDPMs). Our motivation for choosing diffusion models over other generative models comes from the flexible internal structure of diffusion models. Since each sampling step in the DDPM follows a Gaussian distribution, we show that there exists a closed-form solution for generating an image given various constraints. Our method can unite multiple diffusion models trained on multiple sub-tasks and conquer the combined task through our proposed sampling strategy. We also introduce a novel reliability parameter that allows using different off-the-shelf diffusion models trained across various datasets during sampling time alone to guide it to the desired outcome satisfying multiple constraints. We perform experiments on various standard multimodal tasks to demonstrate the effectiveness of our approach. More details can be found in https://nithin-gk.github.io/projectpages/Multidiff/index.html

Denoising diffusion probabilistic models (DDPMs) (Ho et al. 2020) have shown impressive results on image and waveform generation in continuous state spaces. Here, we introduce Discrete Denoising Diffusion Probabilistic Models (D3PMs), diffusion-like generative models for discrete data that generalize the multinomial diffusion model of Hoogeboom et al. 2021, by going beyond corruption processes with uniform transition probabilities. This includes corruption with transition matrices that mimic Gaussian kernels in continuous space, matrices based on nearest neighbors in embedding space, and matrices that introduce absorbing states. The third allows us to draw a connection between diffusion models and autoregressive and mask-based generative models. We show that the choice of transition matrix is an important design decision that leads to improved results in image and text domains. We also introduce a new loss function that combines the variational lower bound with an auxiliary cross entropy loss. For text, this model class achieves strong results on character-level text generation while scaling to large vocabularies on LM1B. On the image dataset CIFAR-10, our models approach the sample quality and exceed the log-likelihood of the continuous-space DDPM model.

Scaling laws offer valuable insights into the design of time series foundation models (TSFMs). However, previous research has largely focused on the scaling laws of TSFMs for in-distribution (ID) data, leaving their out-of-distribution (OOD) scaling behavior and the influence of model architectures less explored. In this work, we examine two common TSFM architectures, encoder-only and decoder-only Transformers, and investigate their scaling behavior on both ID and OOD data. These models are trained and evaluated across varying parameter counts, compute budgets, and dataset sizes. Our experiments reveal that the log-likelihood loss of TSFMs exhibits similar scaling behavior in both OOD and ID settings. We further compare the scaling properties across different architectures, incorporating two state-of-the-art TSFMs as case studies, showing that model architecture plays a significant role in scaling. The encoder-only Transformers demonstrate better scalability than the decoder-only Transformers, while the architectural enhancements in the two advanced TSFMs primarily improve ID performance but reduce OOD scalability. While scaling up TSFMs is expected to drive performance breakthroughs, the lack of a comprehensive understanding of TSFM scaling laws has hindered the development of a robust framework to guide model scaling. We fill this gap in this work by synthesizing our findings and providing practical guidelines for designing and scaling larger TSFMs with enhanced model capabilities.

Building on the success of diffusion models in visual generation, flow-based models reemerge as another prominent family of generative models that have achieved competitive or better performance in terms of both visual quality and inference speed. By learning the velocity field through flow-matching, flow-based models tend to produce a straighter sampling trajectory, which is advantageous during the sampling process. However, unlike diffusion models for which fast samplers are well-developed, efficient sampling of flow-based generative models has been rarely explored. In this paper, we propose a framework called FlowTurbo to accelerate the sampling of flow-based models while still enhancing the sampling quality. Our primary observation is that the velocity predictor's outputs in the flow-based models will become stable during the sampling, enabling the estimation of velocity via a lightweight velocity refiner. Additionally, we introduce several techniques including a pseudo corrector and sample-aware compilation to further reduce inference time. Since FlowTurbo does not change the multi-step sampling paradigm, it can be effectively applied for various tasks such as image editing, inpainting, etc. By integrating FlowTurbo into different flow-based models, we obtain an acceleration ratio of 53.1%sim58.3% on class-conditional generation and 29.8%sim38.5% on text-to-image generation. Notably, FlowTurbo reaches an FID of 2.12 on ImageNet with 100 (ms / img) and FID of 3.93 with 38 (ms / img), achieving the real-time image generation and establishing the new state-of-the-art. Code is available at https://github.com/shiml20/FlowTurbo.

Given an unconditional generative model and a predictor for a target property (e.g., a classifier), the goal of training-free guidance is to generate samples with desirable target properties without additional training. As a highly efficient technique for steering generative models toward flexible outcomes, training-free guidance has gained increasing attention in diffusion models. However, existing methods only handle data in continuous spaces, while many scientific applications involve both continuous and discrete data (referred to as multimodality). Another emerging trend is the growing use of the simple and general flow matching framework in building generative foundation models, where guided generation remains under-explored. To address this, we introduce TFG-Flow, a novel training-free guidance method for multimodal generative flow. TFG-Flow addresses the curse-of-dimensionality while maintaining the property of unbiased sampling in guiding discrete variables. We validate TFG-Flow on four molecular design tasks and show that TFG-Flow has great potential in drug design by generating molecules with desired properties.

We introduce the Series-Parallel Workflow Decomposition (SP\-WD) heuristic algorithm for the Workflow Scheduling Problem (WSP) decomposition. We demonstrate that the SPWD algorithm facilitates the scheduling of large WSP instances with the hybrid D-Wave Constrained Quadratic Model solver, enabling the scheduling of instances that would otherwise exceed its capacity limitations. We also describe the accompanying execution environment used to obtain the results of the experiments with real-life workflow instances available in the WfCommons standardization initiative repository.

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.

This work studies discrete diffusion probabilistic models with applications to natural language generation. We derive an alternative yet equivalent formulation of the sampling from discrete diffusion processes and leverage this insight to develop a family of reparameterized discrete diffusion models. The derived generic framework is highly flexible, offers a fresh perspective of the generation process in discrete diffusion models, and features more effective training and decoding techniques. We conduct extensive experiments to evaluate the text generation capability of our model, demonstrating significant improvements over existing diffusion models.

Generative models transform random noise into images; their inversion aims to transform images back to structured noise for recovery and editing. This paper addresses two key tasks: (i) inversion and (ii) editing of a real image using stochastic equivalents of rectified flow models (such as Flux). Although Diffusion Models (DMs) have recently dominated the field of generative modeling for images, their inversion presents faithfulness and editability challenges due to nonlinearities in drift and diffusion. Existing state-of-the-art DM inversion approaches rely on training of additional parameters or test-time optimization of latent variables; both are expensive in practice. Rectified Flows (RFs) offer a promising alternative to diffusion models, yet their inversion has been underexplored. We propose RF inversion using dynamic optimal control derived via a linear quadratic regulator. We prove that the resulting vector field is equivalent to a rectified stochastic differential equation. Additionally, we extend our framework to design a stochastic sampler for Flux. Our inversion method allows for state-of-the-art performance in zero-shot inversion and editing, outperforming prior works in stroke-to-image synthesis and semantic image editing, with large-scale human evaluations confirming user preference.

Forecasting on sparse multivariate time series (MTS) aims to model the predictors of future values of time series given their incomplete past, which is important for many emerging applications. However, most existing methods process MTS's individually, and do not leverage the dynamic distributions underlying the MTS's, leading to sub-optimal results when the sparsity is high. To address this challenge, we propose a novel generative model, which tracks the transition of latent clusters, instead of isolated feature representations, to achieve robust modeling. It is characterized by a newly designed dynamic Gaussian mixture distribution, which captures the dynamics of clustering structures, and is used for emitting timeseries. The generative model is parameterized by neural networks. A structured inference network is also designed for enabling inductive analysis. A gating mechanism is further introduced to dynamically tune the Gaussian mixture distributions. Extensive experimental results on a variety of real-life datasets demonstrate the effectiveness of our method.

Diffusion models have shown great promise for image and video generation, but sampling from state-of-the-art models requires expensive numerical integration of a generative ODE. One approach for tackling this problem is rectified flows, which iteratively learn smooth ODE paths that are less susceptible to truncation error. However, rectified flows still require a relatively large number of function evaluations (NFEs). In this work, we propose improved techniques for training rectified flows, allowing them to compete with knowledge distillation methods even in the low NFE setting. Our main insight is that under realistic settings, a single iteration of the Reflow algorithm for training rectified flows is sufficient to learn nearly straight trajectories; hence, the current practice of using multiple Reflow iterations is unnecessary. We thus propose techniques to improve one-round training of rectified flows, including a U-shaped timestep distribution and LPIPS-Huber premetric. With these techniques, we improve the FID of the previous 2-rectified flow by up to 72% in the 1 NFE setting on CIFAR-10. On ImageNet 64times64, our improved rectified flow outperforms the state-of-the-art distillation methods such as consistency distillation and progressive distillation in both one-step and two-step settings and rivals the performance of improved consistency training (iCT) in FID. Code is available at https://github.com/sangyun884/rfpp.

Data attribution seeks to trace model outputs back to training data. With the recent development of diffusion models, data attribution has become a desired module to properly assign valuations for high-quality or copyrighted training samples, ensuring that data contributors are fairly compensated or credited. Several theoretically motivated methods have been proposed to implement data attribution, in an effort to improve the trade-off between computational scalability and effectiveness. In this work, we conduct extensive experiments and ablation studies on attributing diffusion models, specifically focusing on DDPMs trained on CIFAR-10 and CelebA, as well as a Stable Diffusion model LoRA-finetuned on ArtBench. Intriguingly, we report counter-intuitive observations that theoretically unjustified design choices for attribution empirically outperform previous baselines by a large margin, in terms of both linear datamodeling score and counterfactual evaluation. Our work presents a significantly more efficient approach for attributing diffusion models, while the unexpected findings suggest that at least in non-convex settings, constructions guided by theoretical assumptions may lead to inferior attribution performance. The code is available at https://github.com/sail-sg/D-TRAK.

Continuous-time Consistency Models (CMs) promise efficient few-step generation but face significant challenges with training instability. We argue this instability stems from a fundamental conflict: by training a network to learn only a shortcut across a probability flow, the model loses its grasp on the instantaneous velocity field that defines the flow. Our solution is to explicitly anchor the model in the underlying flow during training. We introduce the Flow-Anchored Consistency Model (FACM), a simple but effective training strategy that uses a Flow Matching (FM) task as an anchor for the primary CM shortcut objective. This Flow-Anchoring approach requires no architectural modifications and is broadly compatible with standard model architectures. By distilling a pre-trained LightningDiT model, our method achieves a state-of-the-art FID of 1.32 with two steps (NFE=2) and 1.76 with just one step (NFE=1) on ImageNet 256x256, significantly outperforming previous methods. This provides a general and effective recipe for building high-performance, few-step generative models. Our code and pretrained models: https://github.com/ali-vilab/FACM.

Building LiDAR generative models holds promise as powerful data priors for restoration, scene manipulation, and scalable simulation in autonomous mobile robots. In recent years, approaches using diffusion models have emerged, significantly improving training stability and generation quality. Despite their success, diffusion models require numerous iterations of running neural networks to generate high-quality samples, making the increasing computational cost a potential barrier for robotics applications. To address this challenge, this paper presents R2Flow, a fast and high-fidelity generative model for LiDAR data. Our method is based on rectified flows that learn straight trajectories, simulating data generation with significantly fewer sampling steps compared to diffusion models. We also propose an efficient Transformer-based model architecture for processing the image representation of LiDAR range and reflectance measurements. Our experiments on unconditional LiDAR data generation using the KITTI-360 dataset demonstrate the effectiveness of our approach in terms of both efficiency and quality.

Generating graph-structured data requires learning the underlying distribution of graphs. Yet, this is a challenging problem, and the previous graph generative methods either fail to capture the permutation-invariance property of graphs or cannot sufficiently model the complex dependency between nodes and edges, which is crucial for generating real-world graphs such as molecules. To overcome such limitations, we propose a novel score-based generative model for graphs with a continuous-time framework. Specifically, we propose a new graph diffusion process that models the joint distribution of the nodes and edges through a system of stochastic differential equations (SDEs). Then, we derive novel score matching objectives tailored for the proposed diffusion process to estimate the gradient of the joint log-density with respect to each component, and introduce a new solver for the system of SDEs to efficiently sample from the reverse diffusion process. We validate our graph generation method on diverse datasets, on which it either achieves significantly superior or competitive performance to the baselines. Further analysis shows that our method is able to generate molecules that lie close to the training distribution yet do not violate the chemical valency rule, demonstrating the effectiveness of the system of SDEs in modeling the node-edge relationships. Our code is available at https://github.com/harryjo97/GDSS.

Diffusion models (DMs) excel in photorealism, image editing, and solving inverse problems, aided by classifier-free guidance and image inversion techniques. However, rectified flow models (RFMs) remain underexplored for these tasks. Existing DM-based methods often require additional training, lack generalization to pretrained latent models, underperform, and demand significant computational resources due to extensive backpropagation through ODE solvers and inversion processes. In this work, we first develop a theoretical and empirical understanding of the vector field dynamics of RFMs in efficiently guiding the denoising trajectory. Our findings reveal that we can navigate the vector field in a deterministic and gradient-free manner. Utilizing this property, we propose FlowChef, which leverages the vector field to steer the denoising trajectory for controlled image generation tasks, facilitated by gradient skipping. FlowChef is a unified framework for controlled image generation that, for the first time, simultaneously addresses classifier guidance, linear inverse problems, and image editing without the need for extra training, inversion, or intensive backpropagation. Finally, we perform extensive evaluations and show that FlowChef significantly outperforms baselines in terms of performance, memory, and time requirements, achieving new state-of-the-art results. Project Page: https://flowchef.github.io.

Diffusion models approximate the denoising distribution as a Gaussian and predict its mean, whereas flow matching models reparameterize the Gaussian mean as flow velocity. However, they underperform in few-step sampling due to discretization error and tend to produce over-saturated colors under classifier-free guidance (CFG). To address these limitations, we propose a novel Gaussian mixture flow matching (GMFlow) model: instead of predicting the mean, GMFlow predicts dynamic Gaussian mixture (GM) parameters to capture a multi-modal flow velocity distribution, which can be learned with a KL divergence loss. We demonstrate that GMFlow generalizes previous diffusion and flow matching models where a single Gaussian is learned with an L_2 denoising loss. For inference, we derive GM-SDE/ODE solvers that leverage analytic denoising distributions and velocity fields for precise few-step sampling. Furthermore, we introduce a novel probabilistic guidance scheme that mitigates the over-saturation issues of CFG and improves image generation quality. Extensive experiments demonstrate that GMFlow consistently outperforms flow matching baselines in generation quality, achieving a Precision of 0.942 with only 6 sampling steps on ImageNet 256times256.

Generating high-dimensional visual modalities is a computationally intensive task. A common solution is progressive generation, where the outputs are synthesized in a coarse-to-fine spectral autoregressive manner. While diffusion models benefit from the coarse-to-fine nature of denoising, explicit multi-stage architectures are rarely adopted. These architectures have increased the complexity of the overall approach, introducing the need for a custom diffusion formulation, decomposition-dependent stage transitions, add-hoc samplers, or a model cascade. Our contribution, Decomposable Flow Matching (DFM), is a simple and effective framework for the progressive generation of visual media. DFM applies Flow Matching independently at each level of a user-defined multi-scale representation (such as Laplacian pyramid). As shown by our experiments, our approach improves visual quality for both images and videos, featuring superior results compared to prior multistage frameworks. On Imagenet-1k 512px, DFM achieves 35.2% improvements in FDD scores over the base architecture and 26.4% over the best-performing baseline, under the same training compute. When applied to finetuning of large models, such as FLUX, DFM shows faster convergence speed to the training distribution. Crucially, all these advantages are achieved with a single model, architectural simplicity, and minimal modifications to existing training pipelines.

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are strictly more expressive efficient than many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show great potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs' connection to the theory of strongly Rayleigh distributions.

Combinatorial optimization (CO) problems are often NP-hard and thus out of reach for exact algorithms, making them a tempting domain to apply machine learning methods. The highly structured constraints in these problems can hinder either optimization or sampling directly in the solution space. On the other hand, GFlowNets have recently emerged as a powerful machinery to efficiently sample from composite unnormalized densities sequentially and have the potential to amortize such solution-searching processes in CO, as well as generate diverse solution candidates. In this paper, we design Markov decision processes (MDPs) for different combinatorial problems and propose to train conditional GFlowNets to sample from the solution space. Efficient training techniques are also developed to benefit long-range credit assignment. Through extensive experiments on a variety of different CO tasks with synthetic and realistic data, we demonstrate that GFlowNet policies can efficiently find high-quality solutions.

Accelerating diffusion model sampling is crucial for efficient AIGC deployment. While diffusion distillation methods -- based on distribution matching and trajectory matching -- reduce sampling to as few as one step, they fall short on complex tasks like text-to-image generation. Few-step generation offers a better balance between speed and quality, but existing approaches face a persistent trade-off: distribution matching lacks flexibility for multi-step sampling, while trajectory matching often yields suboptimal image quality. To bridge this gap, we propose learning few-step diffusion models by Trajectory Distribution Matching (TDM), a unified distillation paradigm that combines the strengths of distribution and trajectory matching. Our method introduces a data-free score distillation objective, aligning the student's trajectory with the teacher's at the distribution level. Further, we develop a sampling-steps-aware objective that decouples learning targets across different steps, enabling more adjustable sampling. This approach supports both deterministic sampling for superior image quality and flexible multi-step adaptation, achieving state-of-the-art performance with remarkable efficiency. Our model, TDM, outperforms existing methods on various backbones, such as SDXL and PixArt-alpha, delivering superior quality and significantly reduced training costs. In particular, our method distills PixArt-alpha into a 4-step generator that outperforms its teacher on real user preference at 1024 resolution. This is accomplished with 500 iterations and 2 A800 hours -- a mere 0.01% of the teacher's training cost. In addition, our proposed TDM can be extended to accelerate text-to-video diffusion. Notably, TDM can outperform its teacher model (CogVideoX-2B) by using only 4 NFE on VBench, improving the total score from 80.91 to 81.65. Project page: https://tdm-t2x.github.io/

Diffusion models are emerging expressive generative models, in which a large number of time steps (inference steps) are required for a single image generation. To accelerate such tedious process, reducing steps uniformly is considered as an undisputed principle of diffusion models. We consider that such a uniform assumption is not the optimal solution in practice; i.e., we can find different optimal time steps for different models. Therefore, we propose to search the optimal time steps sequence and compressed model architecture in a unified framework to achieve effective image generation for diffusion models without any further training. Specifically, we first design a unified search space that consists of all possible time steps and various architectures. Then, a two stage evolutionary algorithm is introduced to find the optimal solution in the designed search space. To further accelerate the search process, we employ FID score between generated and real samples to estimate the performance of the sampled examples. As a result, the proposed method is (i).training-free, obtaining the optimal time steps and model architecture without any training process; (ii). orthogonal to most advanced diffusion samplers and can be integrated to gain better sample quality. (iii). generalized, where the searched time steps and architectures can be directly applied on different diffusion models with the same guidance scale. Experimental results show that our method achieves excellent performance by using only a few time steps, e.g. 17.86 FID score on ImageNet 64 times 64 with only four steps, compared to 138.66 with DDIM. The code is available at https://github.com/lilijiangg/AutoDiffusion.

Score-based Generative Models (SGMs) have achieved state-of-the-art synthesis results on diverse tasks. However, the current design space of the forward diffusion process is largely unexplored and often relies on physical intuition or simplifying assumptions. Leveraging results from the design of scalable Bayesian posterior samplers, we present a complete recipe for constructing forward processes in SGMs, all of which are guaranteed to converge to the target distribution of interest. We show that several existing SGMs can be cast as specific instantiations of this parameterization. Furthermore, building on this recipe, we construct a novel SGM: Phase Space Langevin Diffusion (PSLD), which performs score-based modeling in a space augmented with auxiliary variables akin to a physical phase space. We show that PSLD outperforms competing baselines in terms of sample quality and the speed-vs-quality tradeoff across different samplers on various standard image synthesis benchmarks. Moreover, we show that PSLD achieves sample quality comparable to state-of-the-art SGMs (FID: 2.10 on unconditional CIFAR-10 generation), providing an attractive alternative as an SGM backbone for further development. We will publish our code and model checkpoints for reproducibility at https://github.com/mandt-lab/PSLD.

Conventional federated learning (FL) frameworks follow a server-driven model where the server determines session initiation and client participation, which faces challenges in accommodating clients' asynchronous needs for model updates. We introduce Client-Driven Federated Learning (CDFL), a novel FL framework that puts clients at the driving role. In CDFL, each client independently and asynchronously updates its model by uploading the locally trained model to the server and receiving a customized model tailored to its local task. The server maintains a repository of cluster models, iteratively refining them using received client models. Our framework accommodates complex dynamics in clients' data distributions, characterized by time-varying mixtures of cluster distributions, enabling rapid adaptation to new tasks with superior performance. In contrast to traditional clustered FL protocols that send multiple cluster models to a client to perform distribution estimation, we propose a paradigm that offloads the estimation task to the server and only sends a single model to a client, and novel strategies to improve estimation accuracy. We provide a theoretical analysis of CDFL's convergence. Extensive experiments across various datasets and system settings highlight CDFL's substantial advantages in model performance and computation efficiency over baselines.

Generative models aim to simulate realistic effects of various actions across different contexts, from text generation to visual effects. Despite significant efforts to build real-world simulators, the application of generative models to virtual worlds, like financial markets, remains under-explored. In financial markets, generative models can simulate complex market effects of participants with various behaviors, enabling interaction under different market conditions, and training strategies without financial risk. This simulation relies on the finest structured data in financial market like orders thus building the finest realistic simulation. We propose Large Market Model (LMM), an order-level generative foundation model, for financial market simulation, akin to language modeling in the digital world. Our financial Market Simulation engine (MarS), powered by LMM, addresses the domain-specific need for realistic, interactive and controllable order generation. Key observations include LMM's strong scalability across data size and model complexity, and MarS's robust and practicable realism in controlled generation with market impact. We showcase MarS as a forecast tool, detection system, analysis platform, and agent training environment, thus demonstrating MarS's "paradigm shift" potential for a variety of financial applications. We release the code of MarS at https://github.com/microsoft/MarS/.

Generative flow networks (GFlowNets) are sequential sampling models trained to match a given distribution. GFlowNets have been successfully applied to various structured object generation tasks, sampling a diverse set of high-reward objects quickly. We propose expected flow networks (EFlowNets), which extend GFlowNets to stochastic environments. We show that EFlowNets outperform other GFlowNet formulations in stochastic tasks such as protein design. We then extend the concept of EFlowNets to adversarial environments, proposing adversarial flow networks (AFlowNets) for two-player zero-sum games. We show that AFlowNets learn to find above 80% of optimal moves in Connect-4 via self-play and outperform AlphaZero in tournaments.

Accurate Subseasonal-to-Seasonal (S2S) ocean simulation is critically important for marine research, yet remains challenging due to its substantial thermal inertia and extended time delay. Machine learning (ML)-based models have demonstrated significant advancements in simulation accuracy and computational efficiency compared to traditional numerical methods. Nevertheless, a significant limitation of current ML models for S2S ocean simulation is their inadequate incorporation of physical consistency and the slow-changing properties of the ocean system. In this work, we propose a neural ocean model (NeuralOM) for S2S ocean simulation with a multi-scale interactive graph neural network to emulate diverse physical phenomena associated with ocean systems effectively. Specifically, we propose a multi-stage framework tailored to model the ocean's slowly changing nature. Additionally, we introduce a multi-scale interactive messaging module to capture complex dynamical behaviors, such as gradient changes and multiplicative coupling relationships inherent in ocean dynamics. Extensive experimental evaluations confirm that our proposed NeuralOM outperforms state-of-the-art models in S2S and extreme event simulation. The codes are available at https://github.com/YuanGao-YG/NeuralOM.

Diffusion and flow matching models have significantly advanced media generation, yet their design space is well-explored, somewhat limiting further improvements. Concurrently, autoregressive (AR) models, particularly those generating continuous tokens, have emerged as a promising direction for unifying text and media generation. This paper introduces Transition Matching (TM), a novel discrete-time, continuous-state generative paradigm that unifies and advances both diffusion/flow models and continuous AR generation. TM decomposes complex generation tasks into simpler Markov transitions, allowing for expressive non-deterministic probability transition kernels and arbitrary non-continuous supervision processes, thereby unlocking new flexible design avenues. We explore these choices through three TM variants: (i) Difference Transition Matching (DTM), which generalizes flow matching to discrete-time by directly learning transition probabilities, yielding state-of-the-art image quality and text adherence as well as improved sampling efficiency. (ii) Autoregressive Transition Matching (ARTM) and (iii) Full History Transition Matching (FHTM) are partially and fully causal models, respectively, that generalize continuous AR methods. They achieve continuous causal AR generation quality comparable to non-causal approaches and potentially enable seamless integration with existing AR text generation techniques. Notably, FHTM is the first fully causal model to match or surpass the performance of flow-based methods on text-to-image task in continuous domains. We demonstrate these contributions through a rigorous large-scale comparison of TM variants and relevant baselines, maintaining a fixed architecture, training data, and hyperparameters.

Diffusion models have been widely adopted in image generation, producing higher-quality and more diverse samples than generative adversarial networks (GANs). We introduce a latent diffusion model (LDM) for precipitation nowcasting - short-term forecasting based on the latest observational data. The LDM is more stable and requires less computation to train than GANs, albeit with more computationally expensive generation. We benchmark it against the GAN-based Deep Generative Models of Rainfall (DGMR) and a statistical model, PySTEPS. The LDM produces more accurate precipitation predictions, while the comparisons are more mixed when predicting whether the precipitation exceeds predefined thresholds. The clearest advantage of the LDM is that it generates more diverse predictions than DGMR or PySTEPS. Rank distribution tests indicate that the distribution of samples from the LDM accurately reflects the uncertainty of the predictions. Thus, LDMs are promising for any applications where uncertainty quantification is important, such as weather and climate.

Mixture models are traditionally represented and learned by adding several distributions as components. Allowing mixtures to subtract probability mass or density can drastically reduce the number of components needed to model complex distributions. However, learning such subtractive mixtures while ensuring they still encode a non-negative function is challenging. We investigate how to learn and perform inference on deep subtractive mixtures by squaring them. We do this in the framework of probabilistic circuits, which enable us to represent tensorized mixtures and generalize several other subtractive models. We theoretically prove that the class of squared circuits allowing subtractions can be exponentially more expressive than traditional additive mixtures; and, we empirically show this increased expressiveness on a series of real-world distribution estimation tasks.

Diffusion models have achieved remarkable success across various domains. However, their slow generation speed remains a critical challenge. Existing acceleration methods, while aiming to reduce steps, often compromise sample quality, controllability, or introduce training complexities. Therefore, we propose RayFlow, a novel diffusion framework that addresses these limitations. Unlike previous methods, RayFlow guides each sample along a unique path towards an instance-specific target distribution. This method minimizes sampling steps while preserving generation diversity and stability. Furthermore, we introduce Time Sampler, an importance sampling technique to enhance training efficiency by focusing on crucial timesteps. Extensive experiments demonstrate RayFlow's superiority in generating high-quality images with improved speed, control, and training efficiency compared to existing acceleration techniques.

Normalizing Flows explicitly maximize a full-dimensional likelihood on the training data. However, real data is typically only supported on a lower-dimensional manifold leading the model to expend significant compute on modeling noise. Injective Flows fix this by jointly learning a manifold and the distribution on it. So far, they have been limited by restrictive architectures and/or high computational cost. We lift both constraints by a new efficient estimator for the maximum likelihood loss, compatible with free-form bottleneck architectures. We further show that naively learning both the data manifold and the distribution on it can lead to divergent solutions, and use this insight to motivate a stable maximum likelihood training objective. We perform extensive experiments on toy, tabular and image data, demonstrating the competitive performance of the resulting model.

Diffusion and flow-based models have enabled significant progress in generation tasks across various modalities and have recently found applications in world modeling. However, unlike typical generation tasks that encourage sample diversity, world models entail different sources of uncertainty and require consistent samples aligned with the ground-truth trajectory, which is a limitation we empirically observe in diffusion models. We argue that a key bottleneck in learning consistent diffusion-based world models lies in the suboptimal predictive ability, which we attribute to the entanglement of condition understanding and target denoising within shared architectures and co-training schemes. To address this, we propose Foresight Diffusion (ForeDiff), a diffusion-based world modeling framework that enhances consistency by decoupling condition understanding from target denoising. ForeDiff incorporates a separate deterministic predictive stream to process conditioning inputs independently of the denoising stream, and further leverages a pretrained predictor to extract informative representations that guide generation. Extensive experiments on robot video prediction and scientific spatiotemporal forecasting show that ForeDiff improves both predictive accuracy and sample consistency over strong baselines, offering a promising direction for diffusion-based world models.

Accurate demand estimation is critical for the retail business in guiding the inventory and pricing policies of perishable products. However, it faces fundamental challenges from censored sales data during stockouts, where unobserved demand creates systemic policy biases. Existing datasets lack the temporal resolution and annotations needed to address this censoring effect. To fill this gap, we present FreshRetailNet-50K, the first large-scale benchmark for censored demand estimation. It comprises 50,000 store-product time series of detailed hourly sales data from 898 stores in 18 major cities, encompassing 863 perishable SKUs meticulously annotated for stockout events. The hourly stock status records unique to this dataset, combined with rich contextual covariates, including promotional discounts, precipitation, and temporal features, enable innovative research beyond existing solutions. We demonstrate one such use case of two-stage demand modeling: first, we reconstruct the latent demand during stockouts using precise hourly annotations. We then leverage the recovered demand to train robust demand forecasting models in the second stage. Experimental results show that this approach achieves a 2.73\% improvement in prediction accuracy while reducing the systematic demand underestimation from 7.37\% to near-zero bias. With unprecedented temporal granularity and comprehensive real-world information, FreshRetailNet-50K opens new research directions in demand imputation, perishable inventory optimization, and causal retail analytics. The unique annotation quality and scale of the dataset address long-standing limitations in retail AI, providing immediate solutions and a platform for future methodological innovation. The data (https://huggingface.co/datasets/Dingdong-Inc/FreshRetailNet-50K) and code (https://github.com/Dingdong-Inc/frn-50k-baseline}) are openly released.

A model involving Gaussian processes (GPs) is introduced to simultaneously handle multi-task learning, clustering, and prediction for multiple functional data. This procedure acts as a model-based clustering method for functional data as well as a learning step for subsequent predictions for new tasks. The model is instantiated as a mixture of multi-task GPs with common mean processes. A variational EM algorithm is derived for dealing with the optimisation of the hyper-parameters along with the hyper-posteriors' estimation of latent variables and processes. We establish explicit formulas for integrating the mean processes and the latent clustering variables within a predictive distribution, accounting for uncertainty on both aspects. This distribution is defined as a mixture of cluster-specific GP predictions, which enhances the performances when dealing with group-structured data. The model handles irregular grid of observations and offers different hypotheses on the covariance structure for sharing additional information across tasks. The performances on both clustering and prediction tasks are assessed through various simulated scenarios and real datasets. The overall algorithm, called MagmaClust, is publicly available as an R package.

Recent advancements in graph diffusion models (GDMs) have enabled the synthesis of realistic network structures, yet ensuring fairness in the generated data remains a critical challenge. Existing solutions attempt to mitigate bias by re-training the GDMs with ad-hoc fairness constraints. Conversely, with this work, we propose FAROS, a novel FAir graph geneRatiOn framework leveraging attribute Switching mechanisms and directly running in the generation process of the pre-trained GDM. Technically, our approach works by altering nodes' sensitive attributes during the generation. To this end, FAROS calculates the optimal fraction of switching nodes, and selects the diffusion step to perform the switch by setting tailored multi-criteria constraints to preserve the node-topology profile from the original distribution (a proxy for accuracy) while ensuring the edge independence on the sensitive attributes for the generated graph (a proxy for fairness). Our experiments on benchmark datasets for link prediction demonstrate that the proposed approach effectively reduces fairness discrepancies while maintaining comparable (or even higher) accuracy performance to other similar baselines. Noteworthy, FAROS is also able to strike a better accuracy-fairness trade-off than other competitors in some of the tested settings under the Pareto optimality concept, demonstrating the effectiveness of the imposed multi-criteria constraints.

Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules and constitute an important step towards the modeling of distributions of many interacting molecules.

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller compared to existing techniques.

Flow-based generative models (Dinh et al., 2014) are conceptually attractive due to tractability of the exact log-likelihood, tractability of exact latent-variable inference, and parallelizability of both training and synthesis. In this paper we propose Glow, a simple type of generative flow using an invertible 1x1 convolution. Using our method we demonstrate a significant improvement in log-likelihood on standard benchmarks. Perhaps most strikingly, we demonstrate that a generative model optimized towards the plain log-likelihood objective is capable of efficient realistic-looking synthesis and manipulation of large images. The code for our model is available at https://github.com/openai/glow

We present Neural Autoregressive Distribution Estimation (NADE) models, which are neural network architectures applied to the problem of unsupervised distribution and density estimation. They leverage the probability product rule and a weight sharing scheme inspired from restricted Boltzmann machines, to yield an estimator that is both tractable and has good generalization performance. We discuss how they achieve competitive performance in modeling both binary and real-valued observations. We also present how deep NADE models can be trained to be agnostic to the ordering of input dimensions used by the autoregressive product rule decomposition. Finally, we also show how to exploit the topological structure of pixels in images using a deep convolutional architecture for NADE.

We study the problem of training a flow-based generative model, parametrized by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture. We provide a sharp end-to-end analysis of the problem. First, we provide a tight closed-form characterization of the learnt velocity field, when parametrized by a shallow denoising auto-encoder trained on a finite number n of samples from the target distribution. Building on this analysis, we provide a sharp description of the corresponding generative flow, which pushes the base Gaussian density forward to an approximation of the target density. In particular, we provide closed-form formulae for the distance between the mean of the generated mixture and the mean of the target mixture, which we show decays as Theta_n(1{n}). Finally, this rate is shown to be in fact Bayes-optimal.

When the performance of a machine learning model varies over groups defined by sensitive attributes (e.g., gender or ethnicity), the performance disparity can be expressed in terms of the probability distributions of the input and output variables over each group. In this paper, we exploit this fact to reduce the disparate impact of a fixed classification model over a population of interest. Given a black-box classifier, we aim to eliminate the performance gap by perturbing the distribution of input variables for the disadvantaged group. We refer to the perturbed distribution as a counterfactual distribution, and characterize its properties for common fairness criteria. We introduce a descent algorithm to learn a counterfactual distribution from data. We then discuss how the estimated distribution can be used to build a data preprocessor that can reduce disparate impact without training a new model. We validate our approach through experiments on real-world datasets, showing that it can repair different forms of disparity without a significant drop in accuracy.

Large-scale AI model training divides work across thousands of GPUs, then synchronizes gradients across them at each step. This incurs a significant network burden that only centralized, monolithic clusters can support, driving up infrastructure costs and straining power systems. We propose Decentralized Diffusion Models, a scalable framework for distributing diffusion model training across independent clusters or datacenters by eliminating the dependence on a centralized, high-bandwidth networking fabric. Our method trains a set of expert diffusion models over partitions of the dataset, each in full isolation from one another. At inference time, the experts ensemble through a lightweight router. We show that the ensemble collectively optimizes the same objective as a single model trained over the whole dataset. This means we can divide the training burden among a number of "compute islands," lowering infrastructure costs and improving resilience to localized GPU failures. Decentralized diffusion models empower researchers to take advantage of smaller, more cost-effective and more readily available compute like on-demand GPU nodes rather than central integrated systems. We conduct extensive experiments on ImageNet and LAION Aesthetics, showing that decentralized diffusion models FLOP-for-FLOP outperform standard diffusion models. We finally scale our approach to 24 billion parameters, demonstrating that high-quality diffusion models can now be trained with just eight individual GPU nodes in less than a week.

Diffusion models have shown remarkable performance on many generative tasks. Despite recent success, most diffusion models are restricted in that they only allow linear transformation of the data distribution. In contrast, broader family of transformations can potentially help train generative distributions more efficiently, simplifying the reverse process and closing the gap between the true negative log-likelihood and the variational approximation. In this paper, we present Neural Diffusion Models (NDMs), a generalization of conventional diffusion models that enables defining and learning time-dependent non-linear transformations of data. We show how to optimise NDMs using a variational bound in a simulation-free setting. Moreover, we derive a time-continuous formulation of NDMs, which allows fast and reliable inference using off-the-shelf numerical ODE and SDE solvers. Finally, we demonstrate the utility of NDMs with learnable transformations through experiments on standard image generation benchmarks, including CIFAR-10, downsampled versions of ImageNet and CelebA-HQ. NDMs outperform conventional diffusion models in terms of likelihood and produce high-quality samples.

This work aims to implement Long Short-Term Memory mixture density networks (LSTM-MDNs) for Value-at-Risk forecasting and compare their performance with established models (historical simulation, CMM, and GARCH) using a defined backtesting procedure. The focus was on the neural network's ability to capture volatility clustering and its real-world applicability. Three architectures were tested: a 2-component mixture density network, a regularized 2-component model (Arimond et al., 2020), and a 3-component mixture model, the latter being tested for the first time in Value-at-Risk forecasting. Backtesting was performed on three stock indices (FTSE 100, S&P 500, EURO STOXX 50) over two distinct two-year periods (2017-2018 as a calm period, 2021-2022 as turbulent). Model performance was assessed through unconditional coverage and independence assumption tests. The neural network's ability to handle volatility clustering was validated via correlation analysis and graphical evaluation. Results show limited success for the neural network approach. LSTM-MDNs performed poorly for 2017/2018 but outperformed benchmark models in 2021/2022. The LSTM mechanism allowed the neural network to capture volatility clustering similarly to GARCH models. However, several issues were identified: the need for proper model initialization and reliance on large datasets for effective learning. The findings suggest that while LSTM-MDNs provide adequate risk forecasts, further research and adjustments are necessary for stable performance.

We study the problem of training diffusion models to sample from a distribution with a given unnormalized density or energy function. We benchmark several diffusion-structured inference methods, including simulation-based variational approaches and off-policy methods (continuous generative flow networks). Our results shed light on the relative advantages of existing algorithms while bringing into question some claims from past work. We also propose a novel exploration strategy for off-policy methods, based on local search in the target space with the use of a replay buffer, and show that it improves the quality of samples on a variety of target distributions. Our code for the sampling methods and benchmarks studied is made public at https://github.com/GFNOrg/gfn-diffusion as a base for future work on diffusion models for amortized inference.

The Mixtures-of-Experts (MoE) model is a widespread distributed and integrated learning method for large language models (LLM), which is favored due to its ability to sparsify and expand models efficiently. However, the performance of MoE is limited by load imbalance and high latency of All-To-All communication, along with relatively redundant computation owing to large expert capacity. Load imbalance may result from existing routing policies that consistently tend to select certain experts. The frequent inter-node communication in the All-To-All procedure also significantly prolongs the training time. To alleviate the above performance problems, we propose a novel routing strategy that combines load balance and locality by converting partial inter-node communication to that of intra-node. Notably, we elucidate that there is a minimum threshold for expert capacity, calculated through the maximal angular deviation between the gating weights of the experts and the assigned tokens. We port these modifications on the PanGu-Sigma model based on the MindSpore framework with multi-level routing and conduct experiments on Ascend clusters. The experiment results demonstrate that the proposed LocMoE reduces training time per epoch by 12.68% to 22.24% compared to classical routers, such as hash router and switch router, without impacting the model accuracy.

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.

Machine learning-based unfolding has enabled unbinned and high-dimensional differential cross section measurements. Two main approaches have emerged in this research area: one based on discriminative models and one based on generative models. The main advantage of discriminative models is that they learn a small correction to a starting simulation while generative models scale better to regions of phase space with little data. We propose to use Schroedinger Bridges and diffusion models to create SBUnfold, an unfolding approach that combines the strengths of both discriminative and generative models. The key feature of SBUnfold is that its generative model maps one set of events into another without having to go through a known probability density as is the case for normalizing flows and standard diffusion models. We show that SBUnfold achieves excellent performance compared to state of the art methods on a synthetic Z+jets dataset.