mlsanityData to 7 Oct 2026

Low-dimensional adaptation of diffusion models: Convergence in total variation

arXiv 2501.12982 ↗First posted 22 Jan 2025stat.ML cs.LGcited by 38

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

This paper investigates how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling. Focusing on two mainstream samplers -- the denoising diffusion implicit model (DDIM) and the denoising diffusion probabilistic model (DDPM), we prove that their iteration complexities under exact score functions are at most the order of k/varepsilon (up to log factor), where varepsilon is the precision in total variation distance and k is some intrinsic dimension of the target distribution. We further extend these convergence guarantees to the setting in which the score functions are learned from data rather than known exactly, showing that the convergence performance degrades gracefully under suitable score estimation assumptions. We then show that these assumptions are attainable via kernel-based score estimators with finite-sample guarantees that also adapt to the low-dimensional structure. Our results apply to a broad family of target distributions without requiring smoothness or log-concavity. Our findings provide the first rigorous evidence for the adaptivity of the DDIM-type samplers to unknown low-dimensional structure, and improve over the state-of-the-art DDPM theory regarding total variation convergence.

Key facts. Cited by 38 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 44 of its 83 references that are in the map. First posted on arXiv 22 Jan 2025.

Builds on

44 of 83 references.

Closest ancestors · 38

Same direction, newest first

  1. 2025A solvable model of learning generative diffusion: theory and insights7 Jan
  2. 2025Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance4 Jan
  3. 2024Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure31 Oct
  4. 2024Provable Acceleration for Diffusion Models under Minimal Assumptions30 Oct
  5. 2024Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplers18 Oct
  6. 2024Instance-dependent Convergence Theory for Diffusion Models17 Oct
  7. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesis11 Oct
  8. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Models7 Oct
  9. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions27 Sep
  10. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions27 Sep
  11. 2024What does guidance do? A fine-grained analysis in a simple setting19 Sep
  12. 2024From optimal score matching to optimal sampling11 Sep
26 more
  1. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions4 Sep
  2. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models5 Aug
  3. 2024Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallel3 Jun
  4. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models23 May
  5. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Models15 Apr
  6. 2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization11 Apr
  7. 2024Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory18 Mar
  8. 2024Accelerating Convergence of Score-Based Diffusion Models, Provably6 Mar
  9. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions23 Feb
  10. 2024Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach21 Feb
  11. 2024Towards a mathematical theory for consistency training in diffusion models12 Feb
  12. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial12 Feb
  13. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances31 Jan
  14. 2024Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization28 Jan
  15. 2023Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Models20 Sep
  16. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models15 Jun
  17. 2023Error Bounds for Flow Matching Methods26 May
  18. 2023Improved Convergence of Score-Based Diffusion Models via Prediction-Correction23 May
  19. 2023The probability flow ODE is provably fast19 May
  20. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  21. 2023Diffusion Models are Minimax Optimal Distribution Estimators3 Mar
  22. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  23. 2022DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models2 Nov
  24. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  25. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models31 Aug
  26. 2020Denoising Diffusion Implicit Models6 Oct

Other references · 6

  1. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,539
  2. 2021The Intrinsic Dimension of Images and Its Impact on Learningcited by 307
  3. 2024Optimal score estimation via empirical Bayes smoothingcited by 60
  4. 2023Convergence of flow-based generative models via proximal gradient descent in Wasserstein spacecited by 34
  5. 2024Optimal convex M-estimation via score matchingcited by 9
  6. 2024Shallow Diffuse: Robust and Invisible Watermarking through Low-Dimensional Subspaces in Diffusion Modelscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 16; 2026: 22.

38 papers.

Sort:

  1. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Modelscited by 23
  2. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  3. 2026Efficient Sampling with Discrete Diffusion Models: Sharp and Adaptive Guaranteescited by 19
  4. 2025Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modelingcited by 18
  5. 2025Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Accelerationcited by 17
  6. 2026Confidence-Based Decoding is Provably Efficient for Diffusion Language Modelscited by 17
  7. 2025Faster Diffusion Models via Higher-Order Approximationcited by 16
  8. 2025Optimal Convergence Analysis of DDPM for General Distributionscited by 16
  9. 2025Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradientscited by 15
  10. 2026High-accuracy sampling for diffusion models and log-concave distributionscited by 13
  11. 2025Generalization of Diffusion Models Arises with a Balanced Representation Spacecited by 10
  12. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
26 more
  1. 2026Adaptation to Intrinsic Dependence in Diffusion Language Modelscited by 9
  2. 2025Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Modelscited by 8
  3. 2025Sublinear iterations can suffice even for DDPMscited by 7
  4. 2025Towards a unified framework for guided diffusion modelscited by 7
  5. 2026High-accuracy and dimension-free sampling with diffusionscited by 7
  6. 2026Provable diffusion-based posterior sampling for linear inverse problems via DDIMcited by 4
  7. 2025Dimension-free error estimate for diffusion model and optimal schedulingcited by 3
  8. 2026Provably adaptive sampling with uniform and remasking discrete diffusion modelscited by 3
  9. 2025Improved Sample Complexity For Diffusion Model Training Without Empirical Risk Minimizer Accesscited by 2
  10. 2026Polynomial Convergence of Riemannian Diffusion Modelscited by 2
  11. 2026Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspectivecited by 2
  12. 2026On the Robustness of Langevin Dynamics to Score Function Errorcited by 2
  13. 2026Wasserstein bounds for denoising diffusion probabilistic models via the F"ollmer processcited by 2
  14. 2026Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributionscited by 2
  15. 2025Understanding Generalization in Diffusion Distillation via Probability Flow Distancecited by 1
  16. 2025Non-asymptotic convergence bound of conditional diffusion modelscited by 1
  17. 2026Entropy-Based Dimension-Free Convergence and Loss-Adaptive Schedules for Diffusion Modelscited by 1
  18. 2026Load--Reserve Wasserstein Propagation for Isotropic Diffusion Samplerscited by 1
  19. 2026On the Robustness of Distribution Support under Diffusion Guidancecited by 1
  20. 2026On the Limits of Latent Reuse in Diffusion Modelscited by 1
  21. 2026Diffusion Models for Adaptive Sequential Data Generationcited by 1
  22. 2026Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choicescited by 1
  23. 2026Transformers Learn the Optimal DDPM Denoiser for Multi-Token GMMscited by 0
  24. 2026When Diffusion Model Can Ignore Dimension: An Entropy-Based Theorycited by 0
  25. 2026Universal Local Error and Realized Amplification for the First-Order EDM Predictorcited by 0
  26. 2026How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectracited by 0

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  2. 2025Diffusion Models: A Mathematical Introduction
  3. 2023Interpreting and Improving Diffusion Models from an Optimization Perspective
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  6. 2021Improved Denoising Diffusion Probabilistic Models
  7. 2026From Score Learning to Discretized Sampling: An End-to-End Generalization Analysis of Diffusion Models
  8. 2025Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Bounds