# Minimax Optimality of the Probability Flow ODE for Diffusion Models

arXiv: https://arxiv.org/abs/2503.09583 · HTML: https://mlsanity.com/paper/2503.09583/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 40 of its 79 references that are in the map.
- First posted on arXiv 12 Mar 2025.

## Placed in

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

## Builds on (most cited first, 30 of 40)

- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Diffusion Model-Based Image Editing: A Survey](https://mlsanity.com/paper/2402.17525/) (2024)
- [Generative Modeling with Denoising Auto-Encoders and Langevin Sampling](https://mlsanity.com/paper/2002.00107/) (2020)
- [Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models](https://mlsanity.com/paper/2306.09251/) (2023)
- [Accelerating Convergence of Score-Based Diffusion Models, Provably](https://mlsanity.com/paper/2403.03852/) (2024)
- [An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization](https://mlsanity.com/paper/2404.07771/) (2024)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial](https://mlsanity.com/paper/2402.07487/) (2024)
- [Convergence Analysis of Probability Flow ODE for Score-based Generative Models](https://mlsanity.com/paper/2404.09730/) (2024)
- [Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models](https://mlsanity.com/paper/2405.14861/) (2024)
- [O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions](https://mlsanity.com/paper/2409.18959/) (2024)
- [Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions](https://mlsanity.com/paper/2402.15602/) (2024)
- [Closed-Form Diffusion Models](https://mlsanity.com/paper/2310.12395/) (2023)
- [Linear Convergence of Diffusion Models Under the Manifold Hypothesis](https://mlsanity.com/paper/2410.09046/) (2024)
- [Learning general Gaussian mixtures with efficient score matching](https://mlsanity.com/paper/2404.18893/) (2024)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [Learning Mixtures of Gaussians Using Diffusion Models](https://mlsanity.com/paper/2404.18869/) (2024)
- [Low-dimensional adaptation of diffusion models: Convergence in total variation](https://mlsanity.com/paper/2501.12982/) (2025)

## Built on by (most cited first, 23 of 23)

- [Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixtures](https://mlsanity.com/paper/2504.05300/) (2025)
- [Breaking AR's Sampling Bottleneck: Provable Acceleration via Diffusion Language Models](https://mlsanity.com/paper/2505.21400/) (2025)
- [Confidence-Based Decoding is Provably Efficient for Diffusion Language Models](https://mlsanity.com/paper/2603.22248/) (2026)
- [Faster Diffusion Models via Higher-Order Approximation](https://mlsanity.com/paper/2506.24042/) (2025)
- [Stage-wise Dynamics of Classifier-Free Guidance in Diffusion Models](https://mlsanity.com/paper/2509.22007/) (2025)
- [Adaptation to Intrinsic Dependence in Diffusion Language Models](https://mlsanity.com/paper/2602.20126/) (2026)
- [Provable Efficiency of Guidance in Diffusion Models for General Data Distribution](https://mlsanity.com/paper/2505.01382/) (2025)
- [Towards a unified framework for guided diffusion models](https://mlsanity.com/paper/2512.04985/) (2025)
- [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025)
- [Diffusion Models with Heavy-Tailed Targets: Score Estimation and Sampling Guarantees](https://mlsanity.com/paper/2601.06715/) (2026)
- [Fast Score-Based Sampling via Log-Concave Reductions](https://mlsanity.com/paper/2512.24152/) (2025)
- [Dimension-free error estimate for diffusion model and optimal scheduling](https://mlsanity.com/paper/2512.01820/) (2025)
- [Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributions](https://mlsanity.com/paper/2605.30153/) (2026)
- [From Conditional to Unconditional Independence: Testing Conditional Independence via Transport Maps](https://mlsanity.com/paper/2504.09567/) (2025)
- [Variation-aware Flexible 3D Gaussian Editing](https://mlsanity.com/paper/2602.11638/) (2026)
- [Denoising growth complexity: Data geometry and certified schedules for diffusion sampling](https://mlsanity.com/paper/2607.26285/) (2026)
- [ConvergeFlow: Language Flow with Provable Convergence to Token Embeddings](https://mlsanity.com/paper/2608.23551/) (2026)
- [Why DDIM Hallucinates More Than DDPM: A Theoretical Analysis of Reverse Dynamics](https://mlsanity.com/paper/2605.06831/) (2026)
- [Diffusion Models for Adaptive Sequential Data Generation](https://mlsanity.com/paper/2606.06007/) (2026)
- [Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choices](https://mlsanity.com/paper/2606.23627/) (2026)
- [Robustness of Diffusion Models under Distribution Shift](https://mlsanity.com/paper/2609.27546/) (2026)
- [Learning a Flow to Self-Supervised Representations](https://mlsanity.com/paper/2609.29350/) (2026)
- [Counterfactual Generation via Flow Matching: Coupling-Sensitive End-to-End Rates](https://mlsanity.com/paper/2610.01193/) (2026)

## Abstract (arXiv)

Score-based diffusion models have become a foundational paradigm for modern generative modeling, demonstrating exceptional capability in generating samples from complex high-dimensional distributions. Despite the dominant adoption of probability flow ODE-based samplers in practice due to their superior sampling efficiency and precision, rigorous statistical guarantees for these methods have remained elusive in the literature. This work develops the first end-to-end theoretical framework for deterministic ODE-based samplers that establishes near-minimax optimal guarantees under mild assumptions on target data distributions. Specifically, focusing on subgaussian distributions with β-H"older smooth densities for β≤ 2, we propose a smooth regularized score estimator that simultaneously controls both the L2 score error and the associated mean Jacobian error. Leveraging this estimator within a refined convergence analysis of the ODE-based sampling process, we demonstrate that the resulting sampler achieves the minimax rate in total variation distance, modulo logarithmic factors. Notably, our theory comprehensively accounts for all sources of error in the sampling process and does not require strong structural conditions such as density lower bounds or Lipschitz/smooth scores on target distributions, thereby covering a broad range of practical data distributions.