Vision-Language-Action (VLA) models offer a unified framework for robotic manipulation by integrating visual perception, language understanding, and control generation. Yet existing VLA models still struggle to generalize across diverse tasks, scenes, and camera viewpoints, and often produce coarse or unstable actions. We introduce E0, a continuized discrete diffusion framework that formulates action generation as iterative denoising over quantized action tokens. Compared with continuous diffusion policies, E0 offers two key advantages: (1) discrete action tokens align naturally with the symbolic structure of pretrained VLM/VLA backbones, enabling stronger semantic conditioning; and 2. discrete diffusion matches the true quantized nature of real-world robot control-whose hardware constraints (e.g., encoder resolution, control frequency, actuation latency) inherently discretize continuous signals-and therefore benefits from a Bayes-optimal denoiser that models the correct discrete action distribution, leading to stronger generalization. Compared with discrete autoregressive and mask-based discrete diffusion models, E0 supports a significantly larger and finer-grained action vocabulary and avoids the distributional mismatch introduced by masking-based corruptions-yielding more accurate fine-grained action control. We further introduce a spherical viewpoint perturbation augmentation method to improve robustness to camera shifts without additional data. Experiments on LIBERO, VLABench, and ManiSkill show that E0 achieves state-of-the-art performance across 14 diverse environments, outperforming strong baselines by 10.7% on average. Real-world evaluation on a Franka arm confirms that E0 delivers precise, robust, and transferable manipulation, establishing discrete diffusion as a promising direction for generalizable VLA policy learning.

While diffusion models excel at generating continuous data such as images, adapting them to discrete tasks has relied on indirect approaches that either operate in continuous embedding spaces or use token masking mechanisms, both of which deviate from modeling the true discrete data distribution that can be theoretically guaranteed by Tweedie's formula. We propose in-situ Tweedie Discrete Diffusion (TDD), a framework that performs diffusion guaranteed by Tweedie's formula directly within the discrete one-hot space, hence "in-situ." Unlike prior methods that diffuse continuous embeddings or mask tokens, TDD directly corrupts one-hot vectors with Gaussian noise and performs iterative denoising through a timestep-conditioned cross-entropy objective rather than mean-squared-error reconstruction. At each denoising step, the model predicts class probabilities, applies argmax to obtain discrete predictions, converts them to one-hot vectors, and feeds them into the next iteration with progressively reduced noise. This process naturally unifies discriminative classification and generative modeling under a single framework. Experiments demonstrate that TDD achieves strong performance on both image classification and text generation tasks, with extensive ablation studies confirming the effectiveness of each design component. Our work establishes a principled approach to discrete diffusion that preserves the core characteristics of diffusion models while operating natively in discrete space.

Generative recommendation represents each item as a semantic ID, i.e., a sequence of discrete tokens, and generates the next item through autoregressive decoding. While effective, existing autoregressive models face two intrinsic limitations: (1) unidirectional constraints, where causal attention restricts each token to attend only to its predecessors, hindering global semantic modeling; and (2) error accumulation, where the fixed left-to-right generation order causes prediction errors in early tokens to propagate to the predictions of subsequent token. To address these issues, we propose LLaDA-Rec, a discrete diffusion framework that reformulates recommendation as parallel semantic ID generation. By combining bidirectional attention with the adaptive generation order, the approach models inter-item and intra-item dependencies more effectively and alleviates error accumulation. Specifically, our approach comprises three key designs: (1) a parallel tokenization scheme that produces semantic IDs for bidirectional modeling, addressing the mismatch between residual quantization and bidirectional architectures; (2) two masking mechanisms at the user-history and next-item levels to capture both inter-item sequential dependencies and intra-item semantic relationships; and (3) an adapted beam search strategy for adaptive-order discrete diffusion decoding, resolving the incompatibility of standard beam search with diffusion-based generation. Experiments on three real-world datasets show that LLaDA-Rec consistently outperforms both ID-based and state-of-the-art generative recommenders, establishing discrete diffusion as a new paradigm for generative recommendation.

We study discrete diffusion for language and other categorical data and focus on a common limitation of masked denoisers: reverse transitions typically factorize across positions, which can weaken joint structure and degrade quality in few-step generation. We propose \emph{Latent Discrete Diffusion Models} (LDDMs), which couple a masked discrete diffusion over tokens with a continuous diffusion over latent embeddings. The latent channel provides a softer signal and carries cross-token dependencies that help resolve ambiguities. We present two instantiations: (i) FUJI-LDDMs, which perform fully joint denoising of tokens and latents, and (ii) SEQ-LDDMs, which sequentially resolve the latent and then the discrete chain conditionally on it. For both variants we derive ELBO-style objectives and discuss design choices to learn informative latents yet amenable to diffusoin modeling. In experiments, LDDMs yield improvements on unconditional generation metrics as compared to state-of-the-art masked discrete diffusion baselines, and are effective at lower sampling budgets, where unmasking many tokens per step is desirable.

In this paper we introduce Hierarchical Diffusion Language Models (HDLM) -- a novel family of discrete diffusion models for language modeling. HDLM builds on a hierarchical vocabulary where low-level tokens with detailed semantics are surjectively mapped to high-level tokens with coarse-grained meanings. In the forward process, each token is independently perturbed to its higher-level ancestor with more abstract semantics according to the scheduler, while in the reverse process the model progressively predicts the next, more detailed semantics. Taken together, HDLM provides a general time-varying next semantic scale prediction process for language modeling. We derive closed-form expressions for the diffusion Evidence Lower Bound (ELBO), and show that HDLM can be implemented in a flexible manner while including the existing MDLM as a special case. We also propose practical training techniques based on the insights. Extensive text generation experiments validate the effectiveness of HDLM, which demonstrates consistently lower validation and generative perplexity than baselines.

Audio inpainting seeks to restore missing segments in degraded recordings. Previous diffusion-based methods exhibit impaired performance when the missing region is large. We introduce the first approach that applies discrete diffusion over tokenized music representations from a pre-trained audio tokenizer, enabling stable and semantically coherent restoration of long gaps. Our method further incorporates two training approaches: a derivative-based regularization loss that enforces smooth temporal dynamics, and a span-based absorbing transition that provides structured corruption during diffusion. Experiments on the MusicNet and MAESTRO datasets with gaps up to 750 ms show that our approach consistently outperforms strong baselines across range of gap lengths, for gaps of 150 ms and above. This work advances musical audio restoration and introduces new directions for discrete diffusion model training. Audio examples of our proposed method can be found at https://iftach21.github.io/.

Diffusion language models, especially masked discrete diffusion models, have achieved great success recently. While there are some theoretical and primary empirical results showing the advantages of latent reasoning with looped transformers or continuous chain-of-thoughts, continuous diffusion models typically underperform their discrete counterparts. In this paper, we argue that diffusion language models do not necessarily need to be in the discrete space. In particular, we prove that continuous diffusion models have stronger expressivity than discrete diffusions and looped transformers. We attribute the contradiction between the theoretical expressiveness and empirical performance to their practical trainability: while continuous diffusion provides intermediate supervision that looped transformers lack, they introduce additional difficulty decoding tokens into the discrete token space from the continuous representation space. We therefore propose Coevolutionary Continuous Discrete Diffusion (CCDD), which defines a joint multimodal diffusion process on the union of a continuous representation space and a discrete token space, leveraging a single model to simultaneously denoise in the joint space. By combining two modalities, CCDD is expressive with rich semantics in the latent space, as well as good trainability and sample quality with the help of explicit discrete tokens. We also propose effective architectures and advanced training/sampling techniques for CCDD, which reveals strong empirical performance in extensive language modeling experiments on real-world tasks.

We study the problem of posterior sampling using pretrained discrete diffusion foundation models, aiming to recover images from noisy measurements without retraining task-specific models. While diffusion models have achieved remarkable success in generative modeling, most advances rely on continuous Gaussian diffusion. In contrast, discrete diffusion offers a unified framework for jointly modeling categorical data such as text and images. Beyond unification, discrete diffusion provides faster inference, finer control, and principled training-free Bayesian inference, making it particularly well-suited for posterior sampling. However, existing approaches to discrete diffusion posterior sampling face severe challenges: derivative-free guidance yields sparse signals, continuous relaxations limit applicability, and split Gibbs samplers suffer from the curse of dimensionality. To overcome these limitations, we introduce Anchored Posterior Sampling (APS) for masked diffusion foundation models, built on two key innovations -- quantized expectation for gradient-like guidance in discrete embedding space, and anchored remasking for adaptive decoding. Our approach achieves state-of-the-art performance among discrete diffusion samplers across linear and nonlinear inverse problems on the standard benchmarks. We further demonstrate the benefits of our approach in training-free stylization and text-guided editing.

We study masked discrete diffusion -- a flexible paradigm for text generation in which tokens are progressively corrupted by special mask symbols before being denoised. Although this approach has demonstrated strong empirical performance, its theoretical complexity in high-dimensional settings remains insufficiently understood. Existing analyses largely focus on uniform discrete diffusion, and more recent attempts addressing masked diffusion either (1) overlook widely used Euler samplers, (2) impose restrictive bounded-score assumptions, or (3) fail to showcase the advantages of masked discrete diffusion over its uniform counterpart. To address this gap, we show that Euler samplers can achieve $ε$-accuracy in total variation (TV) with $\tilde{O}(d^{2}ε^{-3/2})$ discrete score evaluations, thereby providing the first rigorous analysis of typical Euler sampler in masked discrete diffusion. We then propose a Mask-Aware Truncated Uniformization (MATU) approach that both removes bounded-score assumptions and preserves unbiased discrete score approximation. By exploiting the property that each token can be unmasked at most once, MATU attains a nearly $ε$-free complexity of $O(d\,\ln d\cdot (1-ε^2))$. This result surpasses existing uniformization methods under uniform discrete diffusion, eliminating the $\ln(1/ε)$ factor and substantially speeding up convergence. Our findings not only provide a rigorous theoretical foundation for masked discrete diffusion, showcasing its practical advantages over uniform diffusion for text generation, but also pave the way for future efforts to analyze diffusion-based language models developed under masking paradigm.

End-to-End (E2E) solutions have emerged as a mainstream approach for autonomous driving systems, with Vision-Language-Action (VLA) models representing a new paradigm that leverages pre-trained multimodal knowledge from Vision-Language Models (VLMs) to interpret and interact with complex real-world environments. However, these methods remain constrained by the limitations of imitation learning, which struggles to inherently encode physical rules during training. Existing approaches often rely on complex rule-based post-refinement, employ reinforcement learning that remains largely limited to simulation, or utilize diffusion guidance that requires computationally expensive gradient calculations. To address these challenges, we introduce ReflectDrive, a novel learning-based framework that integrates a reflection mechanism for safe trajectory generation via discrete diffusion. We first discretize the two-dimensional driving space to construct an action codebook, enabling the use of pre-trained Diffusion Language Models for planning tasks through fine-tuning. Central to our approach is a safety-aware reflection mechanism that performs iterative self-correction without gradient computation. Our method begins with goal-conditioned trajectory generation to model multi-modal driving behaviors. Based on this, we apply local search methods to identify unsafe tokens and determine feasible solutions, which then serve as safe anchors for inpainting-based regeneration. Evaluated on the NA VSIM benchmark, ReflectDrive demonstrates significant advantages in safety-critical trajectory generation, offering a scalable and reliable solution for autonomous driving systems.