# Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data

arXiv: https://arxiv.org/abs/2302.07194 · HTML: https://mlsanity.com/paper/2302.07194/

## Key facts

- Cited by 174 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 34 of its 53 references that are in the map.
- First posted on arXiv 14 Feb 2023.

## Placed in

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

## Builds on (most cited first, 30 of 34)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Plug and Play Language Models: A Simple Approach to Controlled Text Generation](https://mlsanity.com/paper/1912.02164/) (2019)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (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)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [Generative Modeling with Denoising Auto-Encoders and Langevin Sampling](https://mlsanity.com/paper/2002.00107/) (2020)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation](https://mlsanity.com/paper/2106.05527/) (2021)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)

## Built on by (most cited first, 30 of 174)

- [MMaDA: Multimodal Large Diffusion Language Models](https://mlsanity.com/paper/2505.15809/) (2025)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models](https://mlsanity.com/paper/2307.00619/) (2023)
- [Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems](https://mlsanity.com/paper/2303.05754/) (2023)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models](https://mlsanity.com/paper/2306.09251/) (2023)
- [DiffFit: Unlocking Transferability of Large Diffusion Models via Simple Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2304.06648/) (2023)
- [ReconVLA: Reconstructive Vision-Language-Action Model as Effective Robot Perceiver](https://mlsanity.com/paper/2508.10333/) (2025)
- [An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization](https://mlsanity.com/paper/2404.07771/) (2024)
- [Accelerating Diffusion Sampling with Optimized Time Steps](https://mlsanity.com/paper/2402.17376/) (2024)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (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)
- [Gradient Guidance for Diffusion Models: An Optimization Perspective](https://mlsanity.com/paper/2404.14743/) (2024)
- [Rectified Diffusion: Straightness Is Not Your Need in Rectified Flow](https://mlsanity.com/paper/2410.07303/) (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)
- [Reconstructive Visual Instruction Tuning](https://mlsanity.com/paper/2410.09575/) (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)
- [Reward-Directed Conditional Diffusion: Provable Distribution Estimation and Reward Improvement](https://mlsanity.com/paper/2307.07055/) (2023)
- [What does guidance do? A fine-grained analysis in a simple setting](https://mlsanity.com/paper/2409.13074/) (2024)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models](https://mlsanity.com/paper/2403.01639/) (2024)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [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)

## Abstract (arXiv)

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.