# Continuous diffusion for categorical data

arXiv: https://arxiv.org/abs/2211.15089 · HTML: https://mlsanity.com/paper/2211.15089/

## Key facts

- Cited by 214 papers in the map (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 78 of its 110 references that are in the map.
- First posted on arXiv 28 Nov 2022.

## Placed in

- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)

## Builds on (most cited first, 30 of 78)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [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)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [SentencePiece: A simple and language independent subword tokenizer and detokenizer for Neural Text Processing](https://mlsanity.com/paper/1808.06226/) (2018)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [A Call for Clarity in Reporting BLEU Scores](https://mlsanity.com/paper/1804.08771/) (2018)
- [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)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Scaling Autoregressive Models for Content-Rich Text-to-Image Generation](https://mlsanity.com/paper/2206.10789/) (2022)

## Built on by (most cited first, 30 of 214)

- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Dream 7B: Diffusion Large Language Models](https://mlsanity.com/paper/2508.15487/) (2025)
- [Block Diffusion: Interpolating Between Autoregressive and Diffusion Language Models](https://mlsanity.com/paper/2503.09573/) (2025)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation](https://mlsanity.com/paper/2303.00848/) (2023)
- [Scaling Diffusion Language Models via Adaptation from Autoregressive Models](https://mlsanity.com/paper/2410.17891/) (2024)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Scaling up Masked Diffusion Models on Text](https://mlsanity.com/paper/2410.18514/) (2024)
- [Scalable Adaptive Computation for Iterative Generation](https://mlsanity.com/paper/2212.11972/) (2022)
- [Seed Diffusion: A Large-Scale Diffusion Language Model with High-Speed Inference](https://mlsanity.com/paper/2508.02193/) (2025)
- [Protein Design with Guided Discrete Diffusion](https://mlsanity.com/paper/2305.20009/) (2023)
- [A Reparameterized Discrete Diffusion Model for Text Generation](https://mlsanity.com/paper/2302.05737/) (2023)
- [Likelihood-Based Diffusion Language Models](https://mlsanity.com/paper/2305.18619/) (2023)
- [LLaDA-V: Large Language Diffusion Models with Visual Instruction Tuning](https://mlsanity.com/paper/2505.16933/) (2025)
- [Beyond Autoregression: Discrete Diffusion for Complex Reasoning and Planning](https://mlsanity.com/paper/2410.14157/) (2024)
- [The Diffusion Duality](https://mlsanity.com/paper/2506.10892/) (2025)
- [Unlocking Guidance for Discrete State-Space Diffusion and Flow Models](https://mlsanity.com/paper/2406.01572/) (2024)
- [AR-Diffusion: Auto-Regressive Diffusion Model for Text Generation](https://mlsanity.com/paper/2305.09515/) (2023)
- [Latent Diffusion for Language Generation](https://mlsanity.com/paper/2212.09462/) (2022)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Diffusion Language Models Are Versatile Protein Learners](https://mlsanity.com/paper/2402.18567/) (2024)
- [DiffuRec: A Diffusion Model for Sequential Recommendation](https://mlsanity.com/paper/2304.00686/) (2023)
- [Energy-Based Diffusion Language Models for Text Generation](https://mlsanity.com/paper/2410.21357/) (2024)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (2024)
- [A Survey on Latent Reasoning](https://mlsanity.com/paper/2507.06203/) (2025)

## Abstract (arXiv)

Diffusion models have quickly become the go-to paradigm for generative modelling of perceptual signals (such as images and sound) through iterative refinement. Their success hinges on the fact that the underlying physical phenomena are continuous. For inherently discrete and categorical data such as language, various diffusion-inspired alternatives have been proposed. However, the continuous nature of diffusion models conveys many benefits, and in this work we endeavour to preserve it. We propose CDCD, a framework for modelling categorical data with diffusion models that are continuous both in time and input space. We demonstrate its efficacy on several language modelling tasks.