# Theoretical guarantees in KL for Diffusion Flow Matching

arXiv: https://arxiv.org/abs/2409.08311 · HTML: https://mlsanity.com/paper/2409.08311/

## Key facts

- Cited by 14 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 28 of its 48 references that are in the map.
- First posted on arXiv 12 Sep 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, 28 of 28)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [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)
- [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)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speech](https://mlsanity.com/paper/2105.06337/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Non-Denoising Forward-Time Diffusions](https://mlsanity.com/paper/2312.14589/) (2023)
- [Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances](https://mlsanity.com/paper/2401.17958/) (2024)
- [Convergence of Continuous Normalizing Flows for Learning Probability Distributions](https://mlsanity.com/paper/2404.00551/) (2024)
- [Improved Convergence of Score-Based Diffusion Models via Prediction-Correction](https://mlsanity.com/paper/2305.14164/) (2023)

## Built on by (most cited first, 14 of 14)

- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [CoD: A Diffusion Foundation Model for Image Compression](https://mlsanity.com/paper/2511.18706/) (2025)
- [Non-asymptotic error bounds for probability flow ODEs under weak log-concavity](https://mlsanity.com/paper/2510.17608/) (2025)
- [On Flow Matching KL Divergence](https://mlsanity.com/paper/2511.05480/) (2025)
- [Local Flow Matching Generative Models](https://mlsanity.com/paper/2410.02548/) (2024)
- [Elucidating Rectified Flow with Deterministic Sampler: Polynomial Discretization Complexity for Multi and One-step Models](https://mlsanity.com/paper/2508.08735/) (2025)
- [Exponential Convergence Guarantees for Iterative Markovian Fitting](https://mlsanity.com/paper/2510.20871/) (2025)
- [Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training](https://mlsanity.com/paper/2605.13175/) (2026)
- [Quantile-Coupled Flow Matching for Distributional Reinforcement Learning](https://mlsanity.com/paper/2605.08515/) (2026)
- [Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions](https://mlsanity.com/paper/2605.08882/) (2026)
- [Negative-Guided Subject Fidelity Optimization for Zero-Shot Subject-Driven Generation](https://mlsanity.com/paper/2506.03621/) (2025)
- [Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees](https://mlsanity.com/paper/2606.16610/) (2026)
- [Population-Level Generative Modeling for Ranking Data](https://mlsanity.com/paper/2608.08422/) (2026)
- [Bridge Graphical Models: Coupling, Projection, and Current-Preserving Dynamics for Generative Modeling](https://mlsanity.com/paper/2608.19144/) (2026)

## Abstract (arXiv)

Flow Matching (FM) (also referred to as stochastic interpolants or rectified flows) stands out as a class of generative models that aims to bridge in finite time the target distribution nu^star with an auxiliary distribution μ, leveraging a fixed coupling π and a bridge which can either be deterministic or stochastic. These two ingredients define a path measure which can then be approximated by learning the drift of its Markovian projection. The main contribution of this paper is to provide relatively mild assumptions on nu^star, μ and π to obtain non-asymptotics guarantees for Diffusion Flow Matching (DFM) models using as bridge the conditional distribution associated with the Brownian motion. More precisely, we establish bounds on the Kullback-Leibler divergence between the target distribution and the one generated by such DFM models under moment conditions on the score of nu^star, μ and π, and a standard L2-drift-approximation error assumption.