Diffusion models have become prevalent in generative modeling due to their ability to sample from complex distributions. To improve the quality of generated samples and their compliance with user requirements, two commonly used methods are: (i) Alignment, which involves finetuning a diffusion model to align it with a reward; and (ii) Composition, which combines several pretrained diffusion models together, each emphasizing a desirable attribute in the generated outputs. However, trade-offs often arise when optimizing for multiple rewards or combining multiple models, as they can often represent competing properties. Existing methods cannot guarantee that the resulting model faithfully generates samples with all the desired properties. To address this gap, we propose a constrained optimization framework that unifies alignment and composition of diffusion models by enforcing that the aligned model satisfies reward constraints and/or remains close to each pretrained model. We provide a theoretical characterization of the solutions to the constrained alignment and composition problems and develop a Lagrangian-based primal-dual training algorithm to approximate these solutions. Empirically, we demonstrate our proposed approach in image generation, applying it to alignment and composition, and show that our aligned or composed model satisfies constraints effectively. Our implementation can be found at: https://github.com/shervinkhalafi/constrained_comp_align.
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Applications · Robotics
End-to-end learning has emerged as a transformative paradigm for autonomous driving. However, the inherently multimodal nature of driving behaviors remains a fundamental challenge to robust deployment. We propose DiffE2E, a diffusion-based end-to-end autonomous driving framework. The architecture first performs multi-scale alignment of perception features from multiple sensors via a hierarchical bidirectional cross-attention mechanism. Subsequently, we design a hybrid diffusion-regression-classification decoder based on the Transformer architecture, adopting a collaborative training paradigm to seamlessly fuse the strengths of diffusion and explicit strategies. DiffE2E conducts structured modeling in the latent space: diffusion captures the multimodal distribution of future trajectories, while regression and classification act as explicit strategies to precisely model key control variables such as velocity, enhancing both the precision and controllability of the model. A global condition integration module further enables deep fusion of perception features with high-level goals, significantly improving the quality of trajectory generation. The subsequent cross-attention mechanism facilitates efficient interaction between integrated features and hybrid latent variables, promoting joint optimization of diffusion and explicit strategies for structured output generation and thereby yielding more robust control. Experimental results demonstrate that DiffE2E achieves state-of-the-art performance on both CARLA closed-loop benchmarks and NAVSIM evaluations. The proposed unified framework that integrates diffusion and explicit strategies provides a generalizable paradigm for hybrid action representation and shows substantial potential for extension to broader domains, including embodied intelligence.
Deep Learning · Generative Models and Autoencoders
Recent masked diffusion models (MDMs) have shown competitive performance compared to autoregressive models (ARMs) for language modeling. While most literature has focused on performance enhancing sampling procedures, efficient sampling from MDMs has been scarcely explored. We make the observation that often a given sequence of partially masked tokens determines the values of multiple unknown tokens deterministically, meaning that a single prediction of a masked model holds additional information unused by standard sampling procedures. Based on this observation, we introduce *EB-Sampler*, a simple drop-in replacement for existing samplers, utilizing an **E**ntropy **B**ounded unmasking procedure that dynamically unmasks multiple tokens in one function evaluation with predefined approximate error tolerance. We formulate the EB-Sampler as part of a broad family of adaptive samplers for which we provide an error analysis that motivates our algorithmic choices. EB-Sampler accelerates sampling from current state of the art MDMs by roughly 2-3x on standard coding and math reasoning benchmarks without loss in performance. We also validate the same procedure works well on smaller reasoning tasks including maze navigation and sudoku, tasks ARMs often struggle with.
Deep Learning · Generative Models and Autoencoders
Autoregressive (AR) models have long dominated the landscape of large language models, driving progress across a wide range of tasks. Recently, diffusion-based language models have emerged as a promising alternative, though their advantages over AR models remain underexplored. In this paper, we systematically study masked diffusion models in data-constrained settings—where training involves repeated passes over limited data—and find that they significantly outperform AR models when compute is abundant but data is scarce. Diffusion models make better use of repeated data, achieving lower validation loss and superior downstream performance. We find new scaling laws for diffusion models and derive a closed-form expression for the critical compute threshold at which diffusion begins to outperform AR. Finally, we explain why diffusion models excel in this regime: their randomized masking objective implicitly trains over a rich distribution of token orderings, acting as an implicit data augmentation that AR’s fixed left-to-right factorization lacks. Our results suggest that when data, not compute, is the bottleneck, diffusion models offer a compelling alternative to the standard AR paradigm. Our code is available at: https://diffusion-scaling.github.io
Computer Vision · Image and Video Generation
We introduce LoRAShop, the first framework for multi-concept image generation and editing with LoRA models. LoRAShop builds on a key observation about the feature interaction patterns inside Flux-style diffusion transformers: concept-specific transformer features activate spatially coherent regions early in the denoising process. We harness this observation to derive a disentangled latent mask for each concept in a prior forward pass and blend the corresponding LoRA weights only within regions bounding the concepts to be personalized. The resulting edits seamlessly integrate multiple subjects or styles into the original scene while preserving global context, lighting, and fine details. Our experiments demonstrate that LoRAShop delivers better identity preservation compared to baselines. By eliminating retraining and external constraints, LoRAShop turns personalized diffusion models into a practical `photoshop-with-LoRAs' tool and opens new avenues for compositional visual storytelling and rapid creative iteration.
Computer Vision · Image and Video Generation
Diffusion Transformers have emerged as the foundation for vision generative models, but their scalability is limited by the high cost of hyperparameter (HP) tuning at large scales. Recently, Maximal Update Parametrization ($\mu$P) was proposed for vanilla Transformers, which enables stable HP transfer from small to large language models, and dramatically reduces tuning costs. However, it remains unclear whether $\mu$P of vanilla Transformers extends to diffusion Transformers, which differ architecturally and objectively. In this work, we generalize $\mu$P to diffusion Transformers and validate its effectiveness through large-scale experiments. First, we rigorously prove that $\mu$P of mainstream diffusion Transformers, including DiT, U-ViT, PixArt-$\alpha$, and MMDiT, aligns with that of the vanilla Transformer, enabling the direct application of existing $\mu$P methodologies. Leveraging this result, we systematically demonstrate that DiT-$\mu$P enjoys robust HP transferability. Notably, DiT-XL-2-$\mu$P with transferred learning rate achieves 2.9$\times$ faster convergence than the original DiT-XL-2. Finally, we validate the effectiveness of $\mu$P on text-to-image generation by scaling PixArt-$\alpha$ from 0.04B to 0.61B and MMDiT from 0.18B to 18B. In both cases, models under $\mu$P outperform their respective baselines while requiring small tuning cost—only 5.5% of one training run for PixArt-$\alpha$ and 3% of consumption by human experts for MMDiT-18B. \textit{These results establish $\mu$P as a principled and efficient framework for scaling diffusion Transformers}.
Deep Learning · Robustness
Deep neural networks remain highly susceptible to adversarial attacks, where small, subtle perturbations to input images may induce misclassification. We propose a novel optimization-based purification framework that directly removes these perturbations by maximizing a Bayesian-inspired objective combining a pretrained diffusion prior with a likelihood term tailored to the adversarial perturbation space. Our method iteratively refines a given input through gradient-based updates of a combined score-based loss to guide the purification process. Unlike existing optimization-based defenses that treat adversarial noise as generic corruption, our approach explicitly integrates the adversarial landscape into the objective. Experiments performed on CIFAR-10 and CIFAR-100 demonstrate strong robust accuracy against a range of common adversarial attacks. Our work offers a principled test-time defense grounded in probabilistic inference using score-based generative models. Our code can be found at \url{https://github.com/rooshenasgroup/aaopt}.
Effective trajectory stitching for long-horizon planning is a significant challenge in robotic decision-making. While diffusion models have shown promise in planning, they are limited to solving tasks similar to those seen in their training data. We propose CompDiffuser, a novel generative approach that can solve new tasks by learning to compositionally stitch together shorter trajectory chunks from previously seen tasks. Our key insight is modeling the trajectory distribution by subdividing it into overlapping chunks and learning their conditional relationships through a single bidirectional diffusion model. This allows information to propagate between segments during generation, ensuring physically consistent connections. We conduct experiments on benchmark tasks of various difficulties, covering different environment sizes, agent state dimension, trajectory types, training data quality, and show that CompDiffuser significantly outperforms existing methods.
Computer Vision · Image and Video Generation
Diffusion Transformer (DiT), a promising diffusion model for visual generation, demonstrates impressive performance but incurs significant computational overhead. Intriguingly, analysis of pre-trained DiT models reveals that global self-attention is often redundant, predominantly capturing local patterns—highlighting the potential for more efficient alternatives. In this paper, we revisit convolution as an alternative building block for constructing efficient and expressive diffusion models. However, naively replacing self-attention with convolution typically results in degraded performance. Our investigations attribute this performance gap to the higher channel redundancy in ConvNets compared to Transformers. To resolve this, we introduce a compact channel attention mechanism that promotes the activation of more diverse channels, thereby enhancing feature diversity. This leads to Diffusion ConvNet (DiCo), a family of diffusion models built entirely from standard ConvNet modules, offering strong generative performance with significant efficiency gains. On class-conditional ImageNet generation benchmarks, DiCo-XL achieves an FID of 2.05 at 256$\times$256 resolution and 2.53 at 512$\times$512, with a **2.7$\times$** and **3.1$\times$** speedup over DiT-XL/2, respectively. Furthermore, experimental results on MS-COCO demonstrate that the purely convolutional DiCo exhibits strong potential for text-to-image generation.
Applications · Robotics
Modern AI systems, especially those interacting with the physical world, increasingly require real-time performance. However, the high latency of state-of-the-art generalist models, including recent vision-language-action models (VLAs), poses a significant challenge. While action chunking has enabled temporal consistency in high-frequency control tasks, it does not fully address the latency problem, leading to pauses or out-of-distribution jerky movements at chunk boundaries. This paper presents a novel inference-time algorithm that enables smooth asynchronous execution of action chunking policies. Our method, real-time chunking (RTC), is applicable to any diffusion- or flow-based VLA out of the box with no retraining. It generates the next action chunk while executing the current one, "freezing" actions guaranteed to execute and "inpainting" the rest. To test RTC, we introduce a new benchmark of 12 highly dynamic tasks in the Kinetix simulator, as well as evaluate 6 challenging real-world bimanual manipulation tasks. Results demonstrate that RTC is fast, performant, and uniquely robust to inference delay, significantly improving task throughput and enabling success in precise tasks --- such as lighting a match --- even in the presence of extreme latency.
Computer Vision · Image and Video Generation
In concept erasure, a model is modified to selectively prevent it from generating a target concept. Despite the rapid development of new methods, it remains unclear how thoroughly these approaches remove the target concept from the model. We begin by proposing two conceptual models for the erasure mechanism in diffusion models: (i) interfering with the model’s internal guidance processes, and (ii) reducing the unconditional likelihood of generating the target concept, potentially removing it entirely. To assess whether a concept has been truly erased from the model, we introduce a comprehensive suite of independent probing techniques: supplying visual context, modifying the diffusion trajectory, applying classifier guidance, and analyzing the model's alternative generations that emerge in place of the erased concept. Our results shed light on the value of exploring concept erasure robustness outside of adversarial text inputs, and emphasize the importance of comprehensive evaluations for erasure in diffusion models.
Probabilistic Methods · Monte Carlo and Sampling Methods
We study the problem of posterior sampling in discrete-state spaces using discrete diffusion models. While posterior sampling methods for continuous diffusion models have achieved remarkable progress, analogous methods for discrete diffusion models remain challenging. In this work, we introduce a principled plug-and-play discrete diffusion posterior sampling algorithm based on split Gibbs sampling, which we call SGDD. Our algorithm enables reward-guided generation and solving inverse problems in discrete-state spaces. We demonstrate the convergence of SGDD to the target posterior distribution and verify this through controlled experiments on synthetic benchmarks. Our method enjoys state-of-the-art posterior sampling performance on a range of benchmarks for discrete data, including DNA sequence design, discrete image inverse problems, and music infilling, achieving more than 30% improved performance compared to existing baselines.
Deep Learning · Generative Models and Autoencoders
Inverse problems, where the goal is to recover an unknown signal from noisy or incomplete measurements, are central to applications in medical imaging, remote sensing, and computational biology. Diffusion models have recently emerged as powerful priors for solving such problems. However, existing methods either rely on projection-based techniques that enforce measurement consistency through heuristic updates, or they approximate the likelihood $p(\boldsymbol{y} \mid \boldsymbol{x})$, often resulting in artifacts and instability under complex or high-noise conditions. To address these limitations, we propose a novel framework called coupled data and measurement space diffusion posterior sampling (C-DPS), which eliminates the need for constraint tuning or likelihood approximation. C-DPS introduces a forward stochastic process in the measurement space $\{\boldsymbol{y}_t\}$, evolving in parallel with the data-space diffusion $\{\boldsymbol{x}_t\}$, which enables the derivation of a closed-form posterior $p(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t, \boldsymbol{y}_{t-1})$. This coupling allows for accurate and recursive sampling based on a well-defined posterior distribution. Empirical results demonstrate that C-DPS consistently outperforms existing baselines, both qualitatively and quantitatively, across multiple inverse problem benchmarks.
Computer Vision · 3D Rendering & Reconstruction
Leveraging diffusion models, 3D LiDAR scene generation has achieved great success in both range-view and voxel-based representations. While recent voxel-based approaches can generate both geometric structures and semantic labels, existing range-view methods are limited to producing unlabeled LiDAR scenes. Relying on pretrained segmentation models to predict the semantic maps often results in suboptimal cross-modal consistency. To address this limitation while preserving the advantages of range-view representations, such as computational efficiency and simplified network design, we propose Spiral, a novel range-view LiDAR diffusion model that simultaneously generates depth, reflectance images, and semantic maps. Furthermore, we introduce novel semantic-aware metrics to evaluate the quality of the generated labeled range-view data. Experiments on SemanticKITTI and nuScenes datasets demonstrate that Spiral achieves state-of-the-art performance with the smallest parameter size, outperforming two-step methods that combine the best available generative and segmentation models. Additionally, we validate that Spiral’s generated range images can be effectively used for synthetic data augmentation in the downstream segmentation training, significantly reducing the labeling effort on LiDAR data.
Probabilistic Methods · Monte Carlo and Sampling Methods
Computational methods for learning to sample from the Boltzmann distribution—where the target distribution is known only up to an unnormalized energy function—have advanced significantly recently. Due to the lack of explicit target samples, however, prior diffusion-based methods, known as _diffusion samplers_, often require importance-weighted estimation or complicated learning processes. Both trade off scalability with extensive evaluations of the energy and model, thereby limiting their practical usage. In this work, we propose **Adjoint Schrödinger Bridge Sampler (ASBS)**, a new diffusion sampler that employs simple and scalable matching-based objectives yet without the need to estimate target samples during training. ASBS is grounded on a mathematical model—the Schrödinger Bridge—which enhances sampling efficiency via kinetic-optimal transportation. Through a new lens of stochastic optimal control theory, we demonstrate how SB-based diffusion samplers can be learned at scale via Adjoint Matching and prove convergence to the global solution. Notably, ASBS generalizes the recent Adjoint Sampling (Havens et al., 2025) to arbitrary source distributions by relaxing the so-called memoryless condition that largely restricts the design space. Through extensive experiments, we demonstrate the effectiveness of ASBS on sampling from classical energy functions, amortized conformer generation, and molecular Boltzmann distributions. Codes are available at https://github.com/facebookresearch/adjoint_samplers
Deep Learning · Generative Models and Autoencoders
Negative guidance -- explicitly suppressing unwanted attributes -- remains a fundamental challenge in diffusion models, particularly in few-step sampling regimes. While Classifier-Free Guidance (CFG) works well in standard settings, it fails under aggressive sampling step compression due to divergent predictions between positive and negative branches. We present Normalized Attention Guidance (NAG), an efficient, training-free mechanism that applies extrapolation in attention space with L1-based normalization and refinement. NAG restores effective negative guidance where CFG collapses while maintaining fidelity. Unlike existing approaches, NAG generalizes across architectures (UNet, DiT), sampling regimes (few-step, multi-step), and modalities (image, video), functioning as a \textit{universal} plug-in with minimal computational overhead. Through extensive experimentation, we demonstrate consistent improvements in text alignment (CLIP Score), fidelity (FID, PFID), and human-perceived quality (ImageReward). Our ablation studies validate each design component, while user studies confirm significant preference for NAG-guided outputs. As a model-agnostic inference-time approach requiring no retraining, NAG provides effortless negative guidance for all modern diffusion frameworks -- pseudocode in the Appendix!
Theory · Learning Theory
We develop an analytical framework for understanding how the learned distribution evolves during diffusion model training. Leveraging the Gaussian equivalence principle, we derived exact solutions for the gradient-flow dynamics of weights in one or two layer linear or linear convolutional denoiser settings with arbitrary data, where linear networks converge along principal components, and convolutional networks converge along Fourier modes. Remarkably, these solutions allow us to derive the generated distribution in closed-form and its KL-divergence through training. These analytical results expose a pronounced \emph{spectral bias}, i.e. for both weights and generated distributions, the convergence time of a mode follows an inverse power law of its variance. Empirical experiments on both Gaussian and natural image datasets demonstrate that the power-law spectral bias—remain robust even when using deeper or convolutional architectures. Our results underscore the importance of the data covariance in dictating the order and rate at which diffusion models learn different modes of the data, providing potential explanations of why earlier stopping could lead to incorrect details in image generative model.
Recent advancements in video diffusion models based on Diffusion Transformers (DiTs) have achieved remarkable success in generating temporally coherent videos. Yet, a fundamental question persists: how do these models internally establish and represent temporal correspondences across frames? We introduce DiffTrack, the first quantitative analysis framework designed to answer this question. DiffTrack constructs a dataset of prompt-generated video with pseudo ground-truth tracking annotations and proposes novel evaluation metrics to systematically analyze how each component within the full 3D attention mechanism of DiTs (e.g., representations, layers, and timesteps) contributes to establishing temporal correspondences. Our analysis reveals that query-key similarities in specific (but not all) layers play a critical role in temporal matching, and that this matching becomes increasingly prominent throughout denoising. We demonstrate practical applications of DiffTrack in zero-shot point tracking, where it achieves state-of-the-art performance compared to existing vision foundation and self-supervised video models. Further, we extend our findings to motion-enhanced video generation with a novel guidance method that improves temporal consistency of generated videos without additional training. We believe our work offers crucial insights into the inner workings of video DiTs and establishes a foundation for further research and applications leveraging their temporal understanding.
Deep Learning · Generative Models and Autoencoders
Consistency models are a class of generative models that enable few-step generation for diffusion and flow matching models. While consistency models have achieved promising results on Euclidean domains like images, their applications to Riemannian manifolds remain challenging due to the curved geometry. In this work, we propose the Riemannian Consistency Model (RCM), which, for the first time, enables few-step consistency modeling while respecting the intrinsic manifold constraint imposed by the Riemannian geometry. Leveraging the covariant derivative and exponential-map-based parameterization, we derive the closed-form solutions for both discrete- and continuous-time training objectives for RCM. We then demonstrate theoretical equivalence between the two variants of RCM: Riemannian consistency distillation (RCD) that relies on a teacher model to approximate the marginal vector field, and Riemannian consistency training (RCT) that utilizes the conditional vector field for training. We further propose a simplified training objective that eliminates the need for the complicated differential calculation. Finally, we provide a unique kinematics perspective for interpreting the RCM objective, offering new theoretical angles. Through extensive experiments, we manifest the superior generative quality of RCM in few-step generation on various non-Euclidean manifolds, including flat-tori, spheres, and the 3D rotation group SO(3), spanning a variety of crucial real-world applications such as RNA and protein generation.
Deep Learning · Generative Models and Autoencoders
Diffusion-based generative models have demonstrated exceptional performance, yet their iterative sampling procedures remain computationally expensive. A prominent strategy to mitigate this cost is *distillation*, with *offline distillation* offering particular advantages in terms of efficiency, modularity, and flexibility. In this work, we identify two key observations that motivate a principled distillation framework: (1) while diffusion models have been viewed through the lens of dynamical systems theory, powerful and underexplored tools can be further leveraged; and (2) diffusion models inherently impose structured, semantically coherent trajectories in latent space. Building on these observations, we introduce the *Koopman Distillation Model* (KDM), a novel offline distillation approach grounded in Koopman theory - a classical framework for representing nonlinear dynamics linearly in a transformed space. KDM encodes noisy inputs into an embedded space where a learned linear operator propagates them forward, followed by a decoder that reconstructs clean samples. This enables single-step generation while preserving semantic fidelity. We provide theoretical justification for our approach: (1) under mild assumptions, the learned diffusion dynamics admit a finite-dimensional Koopman representation; and (2) proximity in the Koopman latent space correlates with semantic similarity in the generated outputs, allowing for effective trajectory alignment. Empirically, KDM achieves state-of-the-art performance across standard *offline distillation* benchmarks - improving FID scores by up to 40% in a single generation step.