# Flow Matching for Count Data

arXiv: https://arxiv.org/abs/2605.07746 · HTML: https://mlsanity.com/paper/2605.07746/

## 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 33 of its 59 references that are in the map.
- First posted on arXiv 8 May 2026.

## 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 33)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (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)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [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)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [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)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [GraphDF: A Discrete Flow Model for Molecular Graph Generation](https://mlsanity.com/paper/2102.01189/) (2021)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Integer Discrete Flows and Lossless Compression](https://mlsanity.com/paper/1905.07376/) (2019)
- [Convergence Analysis of Discrete Diffusion Model: Exact Implementation through Uniformization](https://mlsanity.com/paper/2402.08095/) (2024)
- [scDiffusion: conditional generation of high-quality single-cell data using diffusion model](https://mlsanity.com/paper/2401.03968/) (2024)
- [Beyond Masked and Unmasked: Discrete Diffusion Models via Partial Masking](https://mlsanity.com/paper/2505.18495/) (2025)
- [Learning Discrete Distributions by Dequantization](https://mlsanity.com/paper/2001.11235/) (2020)
- [Learning to Jump: Thinning and Thickening Latent Counts for Generative Modeling](https://mlsanity.com/paper/2305.18375/) (2023)

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

- [Stochastic Flow Map for Count Data](https://mlsanity.com/paper/2609.23290/) (2026)
- [CRNDiff: Count-Native Diffusion Framework via Chemical Reaction Networks](https://mlsanity.com/paper/2609.31149/) (2026)
- [Jumping up and down: Denoiser diffusion models for discrete ordinal data](https://mlsanity.com/paper/2610.02754/) (2026)

## Abstract (arXiv)

High-dimensional count data arise in applications such as single-cell RNA sequencing and neural spike trains, where mappings between distributions across successive batches or time points form critical components of data analysis. The recent success of diffusion- and flow-based deep generative models for images, video, and text motivates extending these ideas to count-valued settings, but many existing methods either treat each count as a categorical state or transform counts into a continuous space, neither of which is natural or efficient when the count range is large. We propose count-FM, a flow-matching framework for count data based on a continuous-time birth-death process with local unit jumps. Count-FM learns marginal transitions efficiently in count space through simulation-free training of conditional transition rates, allowing transport between arbitrary count-distributed source and target populations. In simulation, count-FM variants achieve strong sample quality while using substantially fewer parameters. We further apply count-FM to scRNA-seq and neural spike-train data for unconditional generation, transport, and conditional generation. Across these tasks, count-FM yields improved sample quality, greater modeling efficiency, and interpretable transport paths.