# Improved Sample Complexity Bounds for Diffusion Model Training

arXiv: https://arxiv.org/abs/2311.13745 · HTML: https://mlsanity.com/paper/2311.13745/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 24 of its 27 references that are in the map.
- First posted on arXiv 23 Nov 2023.

## Placed in

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

## Builds on (most cited first, 24 of 24)

- [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)
- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [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)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (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)
- [Generative Modeling with Denoising Auto-Encoders and Langevin Sampling](https://mlsanity.com/paper/2002.00107/) (2020)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Provable benefits of score matching](https://mlsanity.com/paper/2306.01993/) (2023)
- [On Size-Independent Sample Complexity of ReLU Networks](https://mlsanity.com/paper/2306.01992/) (2023)
- [High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors](https://mlsanity.com/paper/2302.02497/) (2023)

## Built on by (most cited first, 14 of 14)

- [Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallel](https://mlsanity.com/paper/2406.00924/) (2024)
- [On the Convergence and Straightness of Rectified Flow](https://mlsanity.com/paper/2410.14949/) (2024)
- [Guiding diffusion models to reconstruct flow fields from sparse data](https://mlsanity.com/paper/2510.19971/) (2025)
- [Conditional diffusions for amortized neural posterior estimation](https://mlsanity.com/paper/2410.19105/) (2024)
- [SORSA: Singular Values and Orthonormal Regularized Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2409.00055/) (2024)
- [Discrete State Diffusion Models: A Sample Complexity Perspective](https://mlsanity.com/paper/2510.10854/) (2025)
- [Fine-Tuning Diffusion Models via Intermediate Distribution Shaping](https://mlsanity.com/paper/2510.02692/) (2025)
- [Regularization can make diffusion models more efficient](https://mlsanity.com/paper/2502.09151/) (2025)
- [Improved Sample Complexity For Diffusion Model Training Without Empirical Risk Minimizer Access](https://mlsanity.com/paper/2505.18344/) (2025)
- [Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching](https://mlsanity.com/paper/2512.01286/) (2025)
- [Query Lower Bounds for Diffusion Sampling](https://mlsanity.com/paper/2604.10857/) (2026)
- [Order-Optimal Sample Complexity of Rectified Flows](https://mlsanity.com/paper/2601.20250/) (2026)
- [Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures](https://mlsanity.com/paper/2606.19894/) (2026)
- [Finite-Sample Guarantees for Fully Discrete Conditional Diffusion Models](https://mlsanity.com/paper/2610.13851/) (2026)

## Abstract (arXiv)

Diffusion models have become the most popular approach to deep generative modeling of images, largely due to their empirical performance and reliability. From a theoretical standpoint, a number of recent works have studied the iteration complexity of sampling, assuming access to an accurate diffusion model. In this work, we focus on understanding the sample complexity of training such a model; how many samples are needed to learn an accurate diffusion model using a sufficiently expressive neural network? Prior work showed bounds polynomial in the dimension, desired Total Variation error, and Wasserstein error. We show an exponential improvement in the dependence on Wasserstein error and depth, along with improved dependencies on other relevant parameters.