# α-Flow: A Unified Framework for Continuous-State Discrete Flow Matching Models

arXiv: https://arxiv.org/abs/2504.10283 · HTML: https://mlsanity.com/paper/2504.10283/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 30 of its 47 references that are in the map.
- First posted on arXiv 14 Apr 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [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)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [SSD-LM: Semi-autoregressive Simplex-based Diffusion Language Model for Text Generation and Modular Control](https://mlsanity.com/paper/2210.17432/) (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)
- [Understanding Hallucinations in Diffusion Models through Mode Interpolation](https://mlsanity.com/paper/2406.09358/) (2024)
- [Dirichlet Diffusion Score Model for Biological Sequence Generation](https://mlsanity.com/paper/2305.10699/) (2023)
- [Fisher Flow Matching for Generative Modeling over Discrete Data](https://mlsanity.com/paper/2405.14664/) (2024)
- [TESS: Text-to-Text Self-Conditioned Simplex Diffusion](https://mlsanity.com/paper/2305.08379/) (2023)
- [DPLM-2: A Multimodal Diffusion Protein Language Model](https://mlsanity.com/paper/2410.13782/) (2024)
- [Equivariant Flow Matching with Hybrid Probability Transport](https://mlsanity.com/paper/2312.07168/) (2023)
- [Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective](https://mlsanity.com/paper/2412.03487/) (2024)
- [Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generation](https://mlsanity.com/paper/2404.19739/) (2024)
- [Categorical Flow Matching on Statistical Manifolds](https://mlsanity.com/paper/2405.16441/) (2024)
- [Protein Discovery with Discrete Walk-Jump Sampling](https://mlsanity.com/paper/2306.12360/) (2023)
- [Full-Atom Peptide Design based on Multi-modal Flow Matching](https://mlsanity.com/paper/2406.00735/) (2024)
- [Generative Modeling of Discrete Joint Distributions by E-Geodesic Flow Matching on Assignment Manifolds](https://mlsanity.com/paper/2402.07846/) (2024)
- [Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds](https://mlsanity.com/paper/2406.18806/) (2024)
- [Generative Assignment Flows for Representing and Learning Joint Distributions of Discrete Data](https://mlsanity.com/paper/2406.04527/) (2024)

## Built on by (most cited first, 18 of 18)

- [LangFlow: Continuous Diffusion Rivals Discrete in Language Modeling](https://mlsanity.com/paper/2604.11748/) (2026)
- [WAM-Flow: Parallel Coarse-to-Fine Motion Planning via Discrete Flow Matching for Autonomous Driving](https://mlsanity.com/paper/2512.06112/) (2025)
- [Flow Matching Meets Biology and Life Science: A Survey](https://mlsanity.com/paper/2507.17731/) (2025)
- [Flow Matching is Adaptive to Manifold Structures](https://mlsanity.com/paper/2602.22486/) (2026)
- [Is There a Better Source Distribution than Gaussian? Exploring Source Distributions for Image Flow Matching](https://mlsanity.com/paper/2512.18184/) (2025)
- [Simplex-to-Euclidean Bijections for Categorical Flow Matching](https://mlsanity.com/paper/2510.27480/) (2025)
- [Bi-Anchor Interpolation Solver for Accelerating Generative Modeling](https://mlsanity.com/paper/2601.21542/) (2026)
- [Discrete Beckmann Transport Models for One-Step Language Modeling and Reasoning](https://mlsanity.com/paper/2609.15903/) (2026)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [Geometry-Aware Image Flow Matching](https://mlsanity.com/paper/2605.25294/) (2026)
- [Variable-Length Generative Protein Design via Generalized Poisson Flow](https://mlsanity.com/paper/2607.09039/) (2026)
- [Expanding Flow Maps](https://mlsanity.com/paper/2607.21585/) (2026)
- [Simplex Diffusion Models](https://mlsanity.com/paper/2609.35553/) (2026)
- [Generative Modeling of Discrete Data Using Geometric Latent Subspaces](https://mlsanity.com/paper/2601.21831/) (2026)
- [Towards Faster Language Model Inference Using Mixture-of-Experts Flow Matching](https://mlsanity.com/paper/2604.15009/) (2026)
- [A Time-Reparameterized Cumulative Intensity Extrapolation Sampler for Discrete Flow Matching](https://mlsanity.com/paper/2606.24140/) (2026)
- [Riemannian Deep Learning: Modules, Networks, and Geometries](https://mlsanity.com/paper/2607.19305/) (2026)
- [Bernoulli Flow Models: Self-Consistent Generative Modeling for Binary Data](https://mlsanity.com/paper/2610.11362/) (2026)

## Abstract (arXiv)

Recent efforts have extended the flow-matching framework to discrete generative modeling. One strand of models directly works with the continuous probabilities instead of discrete tokens, which we colloquially refer to as Continuous-State Discrete Flow Matching (CS-DFM). Existing CS-DFM models differ significantly in their representations and geometric assumptions. This work presents a unified framework for CS-DFM models, under which the existing variants can be understood as operating on different α-representations of probabilities. Building upon the theory of information geometry, we introduce α-Flow, a family of CS-DFM models that adheres to the canonical α-geometry of the statistical manifold, and demonstrate its optimality in minimizing the generalized kinetic energy. Theoretically, we show that the flow matching loss for α-flow establishes a unified variational bound for the discrete negative log-likelihood. We comprehensively evaluate different instantiations of α-flow on various discrete generation domains to demonstrate their effectiveness in discrete generative modeling, including intermediate values whose geometries have never been explored before. α-flow significantly outperforms its discrete-state counterpart in image and protein sequence generation and better captures the entropy in language modeling.