# Nonparametric estimation of a factorizable density using diffusion models

arXiv: https://arxiv.org/abs/2501.01783 · HTML: https://mlsanity.com/paper/2501.01783/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 5 of its 115 references that are in the map.
- First posted on arXiv 3 Jan 2025.

## Placed in

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

## Builds on (most cited first, 5 of 5)

- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [On the rates of convergence for learning with convolutional neural networks](https://mlsanity.com/paper/2403.16459/) (2024)
- [Dimension-independent rates for structured neural density estimation](https://mlsanity.com/paper/2411.15095/) (2024)
- [Structural and Convergence Analysis of Discrete-Time Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2406.01320/) (2024)

## Built on by (most cited first, 8 of 8)

- [Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixtures](https://mlsanity.com/paper/2504.05300/) (2025)
- [Localized Diffusion Models](https://mlsanity.com/paper/2505.04417/) (2025)
- [CINDES: Classification induced neural density estimator and simulator](https://mlsanity.com/paper/2510.00367/) (2025)
- [Reflected diffusion models adapt to low-dimensional data](https://mlsanity.com/paper/2603.24495/) (2026)
- [Model-free filtering in high dimensions via projection and score-based diffusions](https://mlsanity.com/paper/2510.23197/) (2025)
- [Statistical Convergence of Spherical First Hitting Diffusion Models](https://mlsanity.com/paper/2605.07625/) (2026)
- [Sample complexity bounds for categorical Markov random fields via Discrete Diffusions](https://mlsanity.com/paper/2610.02128/) (2026)
- [When do explicit penalties improve flow matching?](https://mlsanity.com/paper/2610.15689/) (2026)

## Abstract (arXiv)

In recent years, diffusion models, and more generally score-based deep generative models, have achieved remarkable success in various applications, including image and audio generation. In this paper, we view diffusion models as an implicit approach to nonparametric density estimation and study them within a statistical framework to analyze their surprising performance. A key challenge in high-dimensional statistical inference is leveraging low-dimensional structures inherent in the data to mitigate the curse of dimensionality. We assume that the underlying density exhibits a low-dimensional structure by factorizing into low-dimensional components, a property common in examples such as Bayesian networks and Markov random fields. Under suitable assumptions, we demonstrate that an implicit density estimator constructed from diffusion models adapts to the factorization structure and achieves the minimax optimal rate with respect to the total variation distance. In constructing the estimator, we design a sparse weight-sharing neural network architecture, where sparsity and weight-sharing are key features of practical architectures such as convolutional neural networks and recurrent neural networks.