Localized Diffusion Models
cs.LG, cs.NA, math.NA, stat.ML
Submitted: 2025-05-07
Updated: 2026-09-23
License: http://creativecommons.org/licenses/by/4.0/
The gist: Diffusion models are state-of-the-art tools for various generative tasks.
Terminology
Abstract
Diffusion models are state-of-the-art tools for various generative tasks. Yet training these models involves estimating high-dimensional score functions, a task that in principle suffers from the curse of dimensionality. It is therefore important to understand how low-dimensional structure in the target distribution can be exploited in these models. Here we consider locality structure, which describes certain sparse conditional dependencies among the target random variables. Given some locality structure, the score function is effectively low-dimensional, so that it can be estimated by a localized neural network with significantly reduced sample complexity. This observation motivates the localized diffusion model, where a localized score matching loss is used to train the score function within a localized hypothesis space. We prove that such localization enables diffusion models to circumvent the curse of dimensionality with dimension-independent error bounds, at the price of additional localization error. Under realistic sample size scaling, we then show both theoretically and numerically that a moderate localization radius can balance the statistical and localization errors, yielding better overall performance. Locality structure also facilitates parallel training, making localized diffusion models potentially more efficient for large-scale applications.
Sources
- Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
- An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization
- Stein's method for marginals on large graphical models
- Learning Mixtures of Gaussians Using Diffusion Models
- Localized Schr\"odinger Bridge Sampler
- An analytic theory of creativity in convolutional diffusion models
- Auto-Encoding Variational Bayes
- Towards a Mechanistic Explanation of Diffusion Model Generalization
- Linear Convergence of Diffusion Models Under the Manifold Hypothesis
- Generalization error bound for denoising score matching under relaxed manifold assumption
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