# Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions

arXiv: https://arxiv.org/abs/2605.08882 · HTML: https://mlsanity.com/paper/2605.08882/

## Key facts

- Cited by 1 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 32 of its 40 references that are in the map.
- First posted on arXiv 9 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 32)

- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [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)
- [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)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (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)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Score-based Continuous-time Discrete Diffusion Models](https://mlsanity.com/paper/2211.16750/) (2022)
- [Concrete Score Matching: Generalized Score Matching for Discrete Data](https://mlsanity.com/paper/2211.00802/) (2022)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Non-Denoising Forward-Time Diffusions](https://mlsanity.com/paper/2312.14589/) (2023)
- [Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective](https://mlsanity.com/paper/2412.03487/) (2024)
- [FUDOKI: Discrete Flow-based Unified Understanding and Generation via Kinetic-Optimal Velocities](https://mlsanity.com/paper/2505.20147/) (2025)
- [Absorb and Converge: Provable Convergence Guarantee for Absorbing Discrete Diffusion Models](https://mlsanity.com/paper/2506.02318/) (2025)
- [Discrete Diffusion Models: Novel Analysis and New Sampler Guarantees](https://mlsanity.com/paper/2509.16756/) (2025)
- [Efficient Sampling with Discrete Diffusion Models: Sharp and Adaptive Guarantees](https://mlsanity.com/paper/2602.15008/) (2026)
- [Theoretical guarantees in KL for Diffusion Flow Matching](https://mlsanity.com/paper/2409.08311/) (2024)
- [A Theoretical Analysis of Discrete Flow Matching Generative Models](https://mlsanity.com/paper/2509.22623/) (2025)
- [Non-Asymptotic Convergence of Discrete Diffusion Models: Masked and Random Walk dynamics](https://mlsanity.com/paper/2512.00580/) (2025)
- [Error Analysis of Discrete Flow with Generator Matching](https://mlsanity.com/paper/2509.21906/) (2025)
- [Sharp Convergence Rates for Masked Diffusion Models](https://mlsanity.com/paper/2602.22505/) (2026)
- [Sampling Binary Data by Denoising through Score Functions](https://mlsanity.com/paper/2502.00557/) (2025)

## Built on by (most cited first, 1 of 1)

- [Parallelism, critical windows, and separations among diffusion language models](https://mlsanity.com/paper/2609.20539/) (2026)

## Abstract (arXiv)

Flow Matching has recently emerged as a popular class of generative models for simulating a target distribution μ1 from samples drawn from a source distribution μ0. This framework relies on a fixed coupling between μ0 and μ1, and on a deterministic or stochastic bridge to define an interpolating process between the two distributions. The time marginals of this process can then be approximately sampled by estimating the transition rates, or more generally the generator, of its Markovian projection. This framework has recently been extended to the case of discrete source and target distributions, under the name Discrete Flow Matching (DFM). However, theoretical guarantees for such models remain scarce. In this paper, we study two DFM models on mathbbZmd = 0,ldots,m-1d, sampled through time discretization, and derive non-asymptotic associated bounds for both of them. In contrast to previous work, we establish non-asymptotic bounds in Kullback--Leibler divergence for the early-stopped version of the target distribution. We also derive explicit convergence guarantees in total variation distance with respect to the true target distribution. Importantly, these bounds rely only on an approximation error assumption, relaxing standard score assumptions used in earlier works, while also yielding improved dependence on the vocabulary size m and the dimension d.