# Optimal Convergence Analysis of DDPM for General Distributions arXiv: https://arxiv.org/abs/2510.27562 · HTML: https://mlsanity.com/paper/2510.27562/ ## Key facts - Cited by 16 papers in the map (data through 14 Oct 2026). - Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. - Builds on 50 of its 58 references that are in the map. - First posted on arXiv 31 Oct 2025. ## Placed in - Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/) ## Builds on (most cited first, 30 of 50) - [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) - [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020) - [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022) - [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022) - [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019) - [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021) - [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022) - [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022) - [Diffusion-LM Improves Controllable Text Generation](https://mlsanity.com/paper/2205.14217/) (2022) - [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022) - [AudioLDM: Text-to-Audio Generation with Latent Diffusion Models](https://mlsanity.com/paper/2301.12503/) (2023) - [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022) - [Phenaki: Variable Length Video Generation From Open Domain Textual Description](https://mlsanity.com/paper/2210.02399/) (2022) - [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) - [Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed](https://mlsanity.com/paper/2101.02388/) (2021) - [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) - [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (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) - [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022) - [DPM-Solver-v3: Improved Diffusion ODE Solver with Empirical Model Statistics](https://mlsanity.com/paper/2310.13268/) (2023) - [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023) - [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) - [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024) - [SA-Solver: Stochastic Adams Solver for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2309.05019/) (2023) - [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024) ## Built on by (most cited first, 16 of 16) - [Confidence-Based Decoding is Provably Efficient for Diffusion Language Models](https://mlsanity.com/paper/2603.22248/) (2026) - [High-accuracy sampling for diffusion models and log-concave distributions](https://mlsanity.com/paper/2602.01338/) (2026) - [Adaptation to Intrinsic Dependence in Diffusion Language Models](https://mlsanity.com/paper/2602.20126/) (2026) - [Towards a unified framework for guided diffusion models](https://mlsanity.com/paper/2512.04985/) (2025) - [Sublinear iterations can suffice even for DDPMs](https://mlsanity.com/paper/2511.04844/) (2025) - [High-accuracy and dimension-free sampling with diffusions](https://mlsanity.com/paper/2601.10708/) (2026) - [Provable diffusion-based posterior sampling for linear inverse problems via DDIM](https://mlsanity.com/paper/2607.19333/) (2026) - [On the Robustness of Langevin Dynamics to Score Function Error](https://mlsanity.com/paper/2603.11319/) (2026) - [Query Lower Bounds for Diffusion Sampling](https://mlsanity.com/paper/2604.10857/) (2026) - [Wasserstein bounds for denoising diffusion probabilistic models via the F"ollmer process](https://mlsanity.com/paper/2605.18069/) (2026) - [ConvergeFlow: Language Flow with Provable Convergence to Token Embeddings](https://mlsanity.com/paper/2608.23551/) (2026) - [Connections between the F"ollmer process and the denoising diffusion probabilistic model](https://mlsanity.com/paper/2605.18040/) (2026) - [The Value of Covariance Matching in Gaussian DDPMs and the Lanczos Sampler](https://mlsanity.com/paper/2605.22723/) (2026) - [Smoothed Score Queries and the Complexity of Sampling](https://mlsanity.com/paper/2605.27769/) (2026) - [First-Order Stationarity of Reverse Diffusions](https://mlsanity.com/paper/2609.31612/) (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 achieved remarkable empirical success in generating high-quality samples from target data distributions. Among them, the Denoising Diffusion Probabilistic Model (DDPM) is one of the most widely used samplers, generating samples via estimated score functions. Despite its empirical success, a tight theoretical understanding of DDPM -- especially its convergence properties -- remains limited. In this paper, we provide a refined convergence analysis of the DDPM sampler and establish near-optimal convergence rates under general distributional assumptions. Specifically, we introduce a relaxed smoothness condition parameterized by a constant L, which is small for many practical distributions (e.g., Gaussian mixture models). We prove that the DDPM sampler with accurate score estimates achieves a convergence rate of widetildeOleft(fracdmind,L2T2right)~in Kullback-Leibler divergence, where d is the data dimension, T is the number of iterations, and widetildeO hides polylogarithmic factors in T. This result substantially improves upon the best-known d2/T2 rate when L < √d. By establishing a matching lower bound, we show that our convergence analysis is tight for a wide array of target distributions. Moreover, it reveals that DDPM and DDIM share the same dependence on d, raising an interesting question of why DDIM often appears empirically faster.