# Dimension-free Score Matching and Time Bootstrapping for Diffusion Models

arXiv: https://arxiv.org/abs/2502.10354 · HTML: https://mlsanity.com/paper/2502.10354/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 12 of its 53 references that are in the map.
- First posted on arXiv 14 Feb 2025.

## Placed in

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

## Builds on (most cited first, 12 of 12)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [SSD-LM: Semi-autoregressive Simplex-based Diffusion Language Model for Text Generation and Modular Control](https://mlsanity.com/paper/2210.17432/) (2022)
- [Generative Modeling with Denoising Auto-Encoders and Langevin Sampling](https://mlsanity.com/paper/2002.00107/) (2020)
- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization](https://mlsanity.com/paper/2401.15604/) (2024)
- [Rates of Convergence in the Central Limit Theorem for Markov Chains, with an Application to TD Learning](https://mlsanity.com/paper/2401.15719/) (2024)
- [Shallow diffusion networks provably learn hidden low-dimensional structure](https://mlsanity.com/paper/2410.11275/) (2024)
- [Glauber Generative Model: Discrete Diffusion Models via Binary Classification](https://mlsanity.com/paper/2405.17035/) (2024)

## Built on by (most cited first, 1 of 1)

- [Fine-Tuning Diffusion Models via Intermediate Distribution Shaping](https://mlsanity.com/paper/2510.02692/) (2025)

## Abstract (arXiv)

Diffusion models generate samples by estimating the score function of the target distribution at various noise levels. The model is trained using samples drawn from the target distribution by progressively adding noise. Previous sample complexity bounds have polynomial dependence on the dimension d, apart from a log(|H|) term, where H is the hypothesis class. In this work, we establish the first (nearly) dimension-free sample complexity bounds, modulo the log(|H|) dependence, for learning these score functions, achieving a double exponential improvement in the dimension over prior results. A key aspect of our analysis is the use of a single function approximator to jointly estimate scores across noise levels, a practical feature that enables generalization across time steps. We introduce a martingale-based error decomposition and sharp variance bounds, enabling efficient learning from dependent data generated by Markov processes, which may be of independent interest. Building on these insights, we propose Bootstrapped Score Matching (BSM), a variance reduction technique that leverages previously learned scores to improve accuracy at higher noise levels. These results provide insights into the efficiency and effectiveness of diffusion models for generative modeling.