# Simplex-to-Euclidean Bijections for Categorical Flow Matching

arXiv: https://arxiv.org/abs/2510.27480 · HTML: https://mlsanity.com/paper/2510.27480/

## Key facts

- Cited by 3 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 35 of its 56 references that are in the map.
- First posted on arXiv 31 Oct 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 35)

- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [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)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [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)
- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Analog Bits: Generating Discrete Data using Diffusion Models with Self-Conditioning](https://mlsanity.com/paper/2208.04202/) (2022)
- [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)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [The Diffusion Duality](https://mlsanity.com/paper/2506.10892/) (2025)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (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)
- [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)
- [Categorical Normalizing Flows via Continuous Transformations](https://mlsanity.com/paper/2006.09790/) (2020)
- [TESS 2: A Large-Scale Generalist Diffusion Language Model](https://mlsanity.com/paper/2502.13917/) (2025)
- [Dependent Multinomial Models Made Easy: Stick Breaking with the P'olya-Gamma Augmentation](https://mlsanity.com/paper/1506.05843/) (2015)
- [Invertible Gaussian Reparameterization: Revisiting the Gumbel-Softmax](https://mlsanity.com/paper/1912.09588/) (2019)
- [Learning Discrete Distributions by Dequantization](https://mlsanity.com/paper/2001.11235/) (2020)

## Built on by (most cited first, 3 of 3)

- [Generative Modeling of Discrete Data Using Geometric Latent Subspaces](https://mlsanity.com/paper/2601.21831/) (2026)
- [Simplex-to-Euclidean Bijection for Conjugate and Calibrated Multiclass Gaussian Process Classification](https://mlsanity.com/paper/2603.16621/) (2026)
- [Logit-Coordinate Generative Models for Mixed Continuous-Categorical Tabular Data](https://mlsanity.com/paper/2607.23348/) (2026)

## Abstract (arXiv)

We propose a method for learning and sampling from probability distributions supported on the simplex. Our approach maps the open simplex to Euclidean space via smooth bijections, leveraging the Aitchison geometry to define the mappings, and supports modeling categorical data by a Dirichlet interpolation that dequantizes discrete observations into continuous ones. This enables density modeling in Euclidean space through the bijection while still allowing exact recovery of the original discrete distribution. Compared to previous methods that operate on the simplex using Riemannian geometry or custom noise processes, our approach works in Euclidean space while respecting the Aitchison geometry, and achieves competitive performance on both synthetic and real-world data sets.