# Error Analysis of Discrete Flow with Generator Matching arXiv: https://arxiv.org/abs/2509.21906 ยท HTML: https://mlsanity.com/paper/2509.21906/ ## 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 38 of its 52 references that are in the map. - First posted on arXiv 26 Sep 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 38) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018) - [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) - [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025) - [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) - [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021) - [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022) - [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022) - [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022) - [Your Absorbing Discrete Diffusion Secretly Models the Conditional Distributions of Clean Data](https://mlsanity.com/paper/2406.03736/) (2024) - [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024) - [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017) - [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) - [LLaDA 1.5: Variance-Reduced Preference Optimization for Large Language Diffusion Models](https://mlsanity.com/paper/2505.19223/) (2025) - [MMaDA: Multimodal Large Diffusion Language Models](https://mlsanity.com/paper/2505.15809/) (2025) - [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023) - [d1: Scaling Reasoning in Diffusion Large Language Models via Reinforcement Learning](https://mlsanity.com/paper/2504.12216/) (2025) - [Score-based Continuous-time Discrete Diffusion Models](https://mlsanity.com/paper/2211.16750/) (2022) - [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023) - [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024) - [DeFoG: Discrete Flow Matching for Graph Generation](https://mlsanity.com/paper/2410.04263/) (2024) - [From Denoising Diffusions to Denoising Markov Models](https://mlsanity.com/paper/2211.03595/) (2022) - [Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective](https://mlsanity.com/paper/2412.03487/) (2024) - [How Discrete and Continuous Diffusion Meet: Comprehensive Analysis of Discrete Diffusion Models via a Stochastic Integral Framework](https://mlsanity.com/paper/2410.03601/) (2024) - [Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithms](https://mlsanity.com/paper/2502.00234/) (2025) ## Built on by (most cited first, 9 of 9) - [Non-Asymptotic Convergence of Discrete Diffusion Models: Masked and Random Walk dynamics](https://mlsanity.com/paper/2512.00580/) (2025) - [Demystifying MaskGIT Sampler and Beyond: Adaptive Order Selection in Masked Diffusion](https://mlsanity.com/paper/2510.04525/) (2025) - [Dimension-free error estimate for diffusion model and optimal scheduling](https://mlsanity.com/paper/2512.01820/) (2025) - [Corrected Samplers for Discrete Flow Models](https://mlsanity.com/paper/2601.22519/) (2026) - [Minimax Optimality of Score-Entropy Discrete Diffusion](https://mlsanity.com/paper/2608.20635/) (2026) - [Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions](https://mlsanity.com/paper/2605.08882/) (2026) - [A Time-Reparameterized Cumulative Intensity Extrapolation Sampler for Discrete Flow Matching](https://mlsanity.com/paper/2606.24140/) (2026) - [Entropy Can Flow, or It Can Guide. Be Entropy. LEDFlow: Introducing Entropy-guided Generation Order into Uniform Discrete Flow](https://mlsanity.com/paper/2609.25131/) (2026) - [Acceleration of Diffusion Language Model through Discrete Average Generator](https://mlsanity.com/paper/2609.38364/) (2026) ## Abstract (arXiv) Discrete flow models offer a powerful framework for learning distributions over discrete state spaces and have demonstrated superior performance compared to the discrete diffusion models. However, their convergence properties and error analysis remain largely unexplored. In this work, we develop a unified framework grounded in stochastic calculus theory to systematically investigate the theoretical properties of discrete flow models. Specifically, by leveraging a Girsanov-type theorem for the path measures of two continuous-time Markov chains (CTMCs), we present a comprehensive error analysis that accounts for both transition rate estimation error and early stopping error. In fact, the estimation error of transition rates has received little attention in existing works. Unlike discrete diffusion models, discrete flow incurs no initialization error caused by truncating the time horizon in the noising process. Building on generator matching and uniformization, we establish non-asymptotic error bounds for distribution estimation without the boundedness condition on oracle transition rates. Furthermore, we derive a faster rate of total variation convergence for the estimated distribution with the boundedness condition, yielding a nearly optimal rate in terms of sample size. Our results provide the first error analysis for discrete flow models. We also investigate model performance under different settings based on simulation results.