# Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributions arXiv: https://arxiv.org/abs/2402.08082 · HTML: https://mlsanity.com/paper/2402.08082/ ## Key facts - Cited by 12 papers in the map (data through 14 Oct 2026). - Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. - Builds on 33 of its 46 references that are in the map. - First posted on arXiv 12 Feb 2024. ## Placed in - Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/) ## Builds on (most cited first, 30 of 33) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021) - [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015) - [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021) - [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022) - [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022) - [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) - [Label-Efficient Semantic Segmentation with Diffusion Models](https://mlsanity.com/paper/2112.03126/) (2021) - [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022) - [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021) - [SegDiff: Image Segmentation with Diffusion Probabilistic Models](https://mlsanity.com/paper/2112.00390/) (2021) - [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022) - [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) - [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022) - [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021) - [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) - [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023) - [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) - [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016) - [Step-unrolled Denoising Autoencoders for Text Generation](https://mlsanity.com/paper/2112.06749/) (2021) - [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020) - [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021) - [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controls](https://mlsanity.com/paper/1607.07819/) (2016) - [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020) - [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020) - [On the ability of neural nets to express distributions](https://mlsanity.com/paper/1702.07028/) (2017) ## Built on by (most cited first, 12 of 12) - [Learning Mixtures of Gaussians Using Diffusion Models](https://mlsanity.com/paper/2404.18869/) (2024) - [Generalization of Diffusion Models Arises with a Balanced Representation Space](https://mlsanity.com/paper/2512.20963/) (2025) - [Wasserstein Bounds for generative diffusion models with Gaussian tail targets](https://mlsanity.com/paper/2412.11251/) (2024) - [Dimension-independent rates for structured neural density estimation](https://mlsanity.com/paper/2411.15095/) (2024) - [Diffusion Models with Heavy-Tailed Targets: Score Estimation and Sampling Guarantees](https://mlsanity.com/paper/2601.06715/) (2026) - [Breaking the curse of dimensionality in structured density estimation](https://mlsanity.com/paper/2410.07685/) (2024) - [Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributions](https://mlsanity.com/paper/2605.30153/) (2026) - [Multi-Subspace Multi-Modal Modeling for Diffusion Models: Estimation, Convergence and Mixture of Experts](https://mlsanity.com/paper/2601.01475/) (2026) - [Load--Reserve Wasserstein Propagation for Isotropic Diffusion Samplers](https://mlsanity.com/paper/2603.19670/) (2026) - [Non-asymptotic Convergence of Stochastic Gradient Descent in Score-based Generative Models](https://mlsanity.com/paper/2607.04775/) (2026) - [Gaussian Process Tilted Nonparametric Density Estimation using Fisher Divergence Score Matching](https://mlsanity.com/paper/2504.03485/) (2025) - [Robustness of Diffusion Models under Distribution Shift](https://mlsanity.com/paper/2609.27546/) (2026) ## Abstract (arXiv) While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.