# Unified Discrete Diffusion for Categorical Data

arXiv: https://arxiv.org/abs/2402.03701 · HTML: https://mlsanity.com/paper/2402.03701/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 23 of its 34 references that are in the map.
- First posted on arXiv 6 Feb 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 23 of 23)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Diffusion-LM Improves Controllable Text Generation](https://mlsanity.com/paper/2205.14217/) (2022)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [Text-to-image Diffusion Models in Generative AI: A Survey](https://mlsanity.com/paper/2303.07909/) (2023)
- [On the Importance of Noise Scheduling for Diffusion Models](https://mlsanity.com/paper/2301.10972/) (2023)
- [Score-based Continuous-time Discrete Diffusion Models](https://mlsanity.com/paper/2211.16750/) (2022)
- [A Reparameterized Discrete Diffusion Model for Text Generation](https://mlsanity.com/paper/2302.05737/) (2023)
- [MuseGAN: Multi-track Sequential Generative Adversarial Networks for Symbolic Music Generation and Accompaniment](https://mlsanity.com/paper/1709.06298/) (2017)
- [Conditional molecular design with deep generative models](https://mlsanity.com/paper/1805.00108/) (2018)
- [Learning to design drug-like molecules in three-dimensional space using deep generative models](https://mlsanity.com/paper/2104.08474/) (2021)

## Built on by (most cited first, 11 of 11)

- [Path Planning for Masked Diffusion Model Sampling](https://mlsanity.com/paper/2502.03540/) (2025)
- [Beyond Autoregression: Fast LLMs via Self-Distillation Through Time](https://mlsanity.com/paper/2410.21035/) (2024)
- [Steering Masked Discrete Diffusion Models via Discrete Denoising Posterior Prediction](https://mlsanity.com/paper/2410.08134/) (2024)
- [Loopholing Discrete Diffusion: Deterministic Bypass of the Sampling Wall](https://mlsanity.com/paper/2510.19304/) (2025)
- [On the Detection of Reviewer-Author Collusion Rings From Paper Bidding](https://mlsanity.com/paper/2402.07860/) (2024)
- [Partition Generative Modeling: Masked Modeling Without Masks](https://mlsanity.com/paper/2505.18883/) (2025)
- [DreamReasoner-8B: Block-Size Curriculum Learning for Diffusion Reasoning Models](https://mlsanity.com/paper/2606.19257/) (2026)
- [Accelerating Discrete Diffusion Models with Parallel-In-Time Sampling](https://mlsanity.com/paper/2607.00773/) (2026)
- [Reaction-conditioned De Novo Enzyme Design with GENzyme](https://mlsanity.com/paper/2411.16694/) (2024)
- [Simplex Relaxation for Discrete Diffusion](https://mlsanity.com/paper/2608.10615/) (2026)
- [MolPIF: A Parameter Interpolation Flow Model for Molecule Generation](https://mlsanity.com/paper/2507.13762/) (2025)

## Abstract (arXiv)

Discrete diffusion models have seen a surge of attention with applications on naturally discrete data such as language and graphs. Although discrete-time discrete diffusion has been established for a while, only recently Campbell et al. (2022) introduced the first framework for continuous-time discrete diffusion. However, their training and sampling processes differ significantly from the discrete-time version, necessitating nontrivial approximations for tractability. In this paper, we first present a series of mathematical simplifications of the variational lower bound that enable more accurate and easy-to-optimize training for discrete diffusion. In addition, we derive a simple formulation for backward denoising that enables exact and accelerated sampling, and importantly, an elegant unification of discrete-time and continuous-time discrete diffusion. Thanks to simpler analytical formulations, both forward and now also backward probabilities can flexibly accommodate any noise distribution, including different noise distributions for multi-element objects. Experiments show that our proposed USD3 (for Unified Simplified Discrete Denoising Diffusion) outperform all SOTA baselines on established datasets. We open-source our unified code at https://github.com/LingxiaoShawn/USD3.