# A Quantitative Approximation Framework for Flow Distillation in Diffusion Models

arXiv: https://arxiv.org/abs/2606.03820 · HTML: https://mlsanity.com/paper/2606.03820/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 51 of its 60 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 51)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [One-step Diffusion with Distribution Matching Distillation](https://mlsanity.com/paper/2311.18828/) (2023)
- [Adversarial Diffusion Distillation](https://mlsanity.com/paper/2311.17042/) (2023)
- [Improved Distribution Matching Distillation for Fast Image Synthesis](https://mlsanity.com/paper/2405.14867/) (2024)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [BOOT: Data-free Distillation of Denoising Diffusion Models with Bootstrapping](https://mlsanity.com/paper/2306.05544/) (2023)
- [Align Your Flow: Scaling Continuous-Time Flow Map Distillation](https://mlsanity.com/paper/2506.14603/) (2025)
- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [What does guidance do? A fine-grained analysis in a simple setting](https://mlsanity.com/paper/2409.13074/) (2024)
- [Critical windows: non-asymptotic theory for feature emergence in diffusion models](https://mlsanity.com/paper/2403.01633/) (2024)
- [Exact Diffusion Inversion via Bi-directional Integration Approximation](https://mlsanity.com/paper/2307.10829/) (2023)
- [Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization](https://mlsanity.com/paper/2401.15604/) (2024)
- [Learning Mixtures of Gaussians Using Diffusion Models](https://mlsanity.com/paper/2404.18869/) (2024)
- [Analysis of learning a flow-based generative model from limited sample complexity](https://mlsanity.com/paper/2310.03575/) (2023)

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

- [Cross-Time Directional Selection in Diffusion Sampling](https://mlsanity.com/paper/2610.05199/) (2026)

## Abstract (arXiv)

We develop a quantitative framework for diffusion distillation by viewing few step sampling as approximation through compositions of learned flow maps. For trajectory distillation of the probability flow ODE, we show that low noise multimodal regimes separate score approximability from dynamical stability: the score remains efficiently approximable, while small local errors may be strongly amplified by stiff flow dynamics. In a Gaussian mixture Ornstein--Uhlenbeck model, we prove time uniform (Lp(pt)) score approximation by ReLU and ReQU networks with explicit polylogarithmic complexity, and derive a computable Lipschitz bound (L(t)) for the flow velocity. The stability factor (expbigl(intst L(u)mathrm dubigr)) can grow exponentially as noise decreases and mixture separation increases. Comparing this certificate with a certified local Lipschitz budget for one step students identifies regimes of direct distillation difficulty, without implying an approximation lower bound. We also show that deep residual compositions control global transport error through propagated local errors, and that equalizing cumulative stability yields an optimal nonuniform segmentation. With eight segments, this grid reduces final mean relative MSE by up to (51.9%) versus uniform grids.