# Generative Modeling of Discrete Joint Distributions by E-Geodesic Flow Matching on Assignment Manifolds

arXiv: https://arxiv.org/abs/2402.07846 · HTML: https://mlsanity.com/paper/2402.07846/

## Key facts

- Cited by 9 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 26 of its 44 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## 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, 26 of 26)

- [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)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [The Cityscapes Dataset for Semantic Urban Scene Understanding](https://mlsanity.com/paper/1604.01685/) (2016)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [Analog Bits: Generating Discrete Data using Diffusion Models with Self-Conditioning](https://mlsanity.com/paper/2208.04202/) (2022)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Integer Discrete Flows and Lossless Compression](https://mlsanity.com/paper/1905.07376/) (2019)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Discrete Flows: Invertible Generative Models of Discrete Data](https://mlsanity.com/paper/1905.10347/) (2019)
- [Moser Flow: Divergence-based Generative Modeling on Manifolds](https://mlsanity.com/paper/2108.08052/) (2021)
- [Matching Normalizing Flows and Probability Paths on Manifolds](https://mlsanity.com/paper/2207.04711/) (2022)
- [Categorical Normalizing Flows via Continuous Transformations](https://mlsanity.com/paper/2006.09790/) (2020)
- [Invertible Gaussian Reparameterization: Revisiting the Gumbel-Softmax](https://mlsanity.com/paper/1912.09588/) (2019)
- [Semi-Discrete Normalizing Flows through Differentiable Tessellation](https://mlsanity.com/paper/2203.06832/) (2022)
- [Assignment Flows for Data Labeling on Graphs: Convergence and Stability](https://mlsanity.com/paper/2002.11571/) (2020)
- [A Geometric Embedding Approach to Multiple Games and Multiple Populations](https://mlsanity.com/paper/2401.05918/) (2024)

## Built on by (most cited first, 9 of 9)

- [Fisher Flow Matching for Generative Modeling over Discrete Data](https://mlsanity.com/paper/2405.14664/) (2024)
- [Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generation](https://mlsanity.com/paper/2404.19739/) (2024)
- [α-Flow: A Unified Framework for Continuous-State Discrete Flow Matching Models](https://mlsanity.com/paper/2504.10283/) (2025)
- [Generative Assignment Flows for Representing and Learning Joint Distributions of Discrete Data](https://mlsanity.com/paper/2406.04527/) (2024)
- [Spherical Flows for Sampling Categorical Data](https://mlsanity.com/paper/2605.05629/) (2026)
- [MolPIF: A Parameter Interpolation Flow Model for Molecule Generation](https://mlsanity.com/paper/2507.13762/) (2025)
- [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)
- [EvoEGF-Mol: Evolving Exponential Geodesic Flow for Structure-based Drug Design](https://mlsanity.com/paper/2601.22466/) (2026)

## Abstract (arXiv)

This paper introduces a novel generative model for discrete distributions based on continuous normalizing flows on the submanifold of factorizing discrete measures. Integration of the flow gradually assigns categories and avoids issues of discretizing the latent continuous model like rounding, sample truncation etc. General non-factorizing discrete distributions capable of representing complex statistical dependencies of structured discrete data, can be approximated by embedding the submanifold into a the meta-simplex of all joint discrete distributions and data-driven averaging. Efficient training of the generative model is demonstrated by matching the flow of geodesics of factorizing discrete distributions. Various experiments underline the approach's broad applicability.