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

arXiv: https://arxiv.org/abs/2501.12982 · HTML: https://mlsanity.com/paper/2501.12982/

## Key facts

- Cited by 39 papers in the map (data through 14 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.

## Placed in

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

## Builds on (most cited first, 30 of 44)

- [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)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (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)
- [Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers](https://mlsanity.com/paper/2303.03384/) (2023)
- [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)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (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)
- [Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances](https://mlsanity.com/paper/2401.17958/) (2024)
- [Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models](https://mlsanity.com/paper/2405.14861/) (2024)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions](https://mlsanity.com/paper/2409.18959/) (2024)
- [What does guidance do? A fine-grained analysis in a simple setting](https://mlsanity.com/paper/2409.13074/) (2024)
- [Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions](https://mlsanity.com/paper/2402.15602/) (2024)
- [Linear Convergence of Diffusion Models Under the Manifold Hypothesis](https://mlsanity.com/paper/2410.09046/) (2024)
- [Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization](https://mlsanity.com/paper/2401.15604/) (2024)
- [Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Models](https://mlsanity.com/paper/2309.11420/) (2023)
- [Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure](https://mlsanity.com/paper/2410.24060/) (2024)

## Built on by (most cited first, 30 of 39)

- [Minimax Optimality of the Probability Flow ODE for Diffusion Models](https://mlsanity.com/paper/2503.09583/) (2025)
- [Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixtures](https://mlsanity.com/paper/2504.05300/) (2025)
- [Efficient Sampling with Discrete Diffusion Models: Sharp and Adaptive Guarantees](https://mlsanity.com/paper/2602.15008/) (2026)
- [Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modeling](https://mlsanity.com/paper/2502.05743/) (2025)
- [Confidence-Based Decoding is Provably Efficient for Diffusion Language Models](https://mlsanity.com/paper/2603.22248/) (2026)
- [Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Acceleration](https://mlsanity.com/paper/2502.04849/) (2025)
- [Faster Diffusion Models via Higher-Order Approximation](https://mlsanity.com/paper/2506.24042/) (2025)
- [Optimal Convergence Analysis of DDPM for General Distributions](https://mlsanity.com/paper/2510.27562/) (2025)
- [High-accuracy sampling for diffusion models and log-concave distributions](https://mlsanity.com/paper/2602.01338/) (2026)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [Generalization of Diffusion Models Arises with a Balanced Representation Space](https://mlsanity.com/paper/2512.20963/) (2025)
- [Adaptation to Intrinsic Dependence in Diffusion Language Models](https://mlsanity.com/paper/2602.20126/) (2026)
- [Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models](https://mlsanity.com/paper/2506.13061/) (2025)
- [Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Models](https://mlsanity.com/paper/2501.18863/) (2025)
- [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)
- [Provably adaptive sampling with uniform and remasking discrete diffusion models](https://mlsanity.com/paper/2608.23554/) (2026)
- [Dimension-free error estimate for diffusion model and optimal scheduling](https://mlsanity.com/paper/2512.01820/) (2025)
- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026)
- [On the Robustness of Langevin Dynamics to Score Function Error](https://mlsanity.com/paper/2603.11319/) (2026)
- [Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributions](https://mlsanity.com/paper/2605.30153/) (2026)
- [Improved Sample Complexity For Diffusion Model Training Without Empirical Risk Minimizer Access](https://mlsanity.com/paper/2505.18344/) (2025)
- [Understanding Generalization in Diffusion Distillation via Probability Flow Distance](https://mlsanity.com/paper/2505.20123/) (2025)
- [Wasserstein bounds for denoising diffusion probabilistic models via the F"ollmer process](https://mlsanity.com/paper/2605.18069/) (2026)
- [Non-asymptotic convergence bound of conditional diffusion models](https://mlsanity.com/paper/2508.10944/) (2025)
- [Entropy-Based Dimension-Free Convergence and Loss-Adaptive Schedules for Diffusion Models](https://mlsanity.com/paper/2601.21943/) (2026)
- [Load--Reserve Wasserstein Propagation for Isotropic Diffusion Samplers](https://mlsanity.com/paper/2603.19670/) (2026)

## 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.