Localized Diffusion Models for High Dimensional Distributions Generation

• URL: http://arxiv.org/abs/2505.04417v1
• Date: Wed, 07 May 2025 13:51:50 GMT
• Title: Localized Diffusion Models for High Dimensional Distributions Generation
• Authors: Georg A. Gottwald, Shuigen Liu, Youssef Marzouk, Sebastian Reich, Xin T. Tong,
• Abstract summary: We study the importance of better understanding and exploiting low-dimensional structure in the target distribution.<n>Under 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.<n>We prove that such localization enables diffusion models to circumvent the curse of dimensionality (CoD)
• Score: 1.677718351174347
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Diffusion models are the state-of-the-art tools for various generative tasks. However, estimating high-dimensional score functions makes them potentially suffer from the curse of dimensionality (CoD). This underscores the importance of better understanding and exploiting low-dimensional structure in the target distribution. In this work, we consider locality structure, which describes sparse dependencies between model components. Under 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 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 CoD, at the price of additional localization error. Under realistic sample size scaling, we show both theoretically and numerically that a moderate localization radius can balance the statistical and localization error, leading to a better overall performance. The localized structure also facilitates parallel training of diffusion models, making it potentially more efficient for large-scale applications.

Related papers

• Distributed Gradient Descent with Many Local Steps in Overparameterized Models [20.560882414631784]
  In distributed training of machine learning models, gradient descent with local iterative steps is a popular method.<n>We try to explain this good performance from a viewpoint of implicit bias in Local Gradient Descent (Local-GD) with a large number of local steps.
  arXiv  Detail & Related papers  (2024-12-10T23:19:40Z)
• Stein's method for marginals on large graphical models [1.8843687952462742]
  We introduce a novel $delta$-locality condition that quantifies the locality in distributions.<n>We show that these methods greatly reduce the sample complexity and computational cost via localized and parallel implementations.
  arXiv  Detail & Related papers  (2024-10-15T16:47:05Z)
• Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models [6.76974373198208]
  We find that the dependency of the error incurred within each denoising step on the ambient dimension $d$ is in general unavoidable.<n>This represents the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures in the target distribution.
  arXiv  Detail & Related papers  (2024-05-23T17:59:10Z)
• Analyzing Neural Network-Based Generative Diffusion Models through Convex Optimization [45.72323731094864]
  We present a theoretical framework to analyze two-layer neural network-based diffusion models. We prove that training shallow neural networks for score prediction can be done by solving a single convex program. Our results provide a precise characterization of what neural network-based diffusion models learn in non-asymptotic settings.
  arXiv  Detail & Related papers  (2024-02-03T00:20:25Z)
• Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution [67.9215891673174]
  We propose score entropy as a novel loss that naturally extends score matching to discrete spaces. We test our Score Entropy Discrete Diffusion models on standard language modeling tasks.
  arXiv  Detail & Related papers  (2023-10-25T17:59:12Z)
• Diffusion Models are Minimax Optimal Distribution Estimators [49.47503258639454]
  We provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling. We show that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates.
  arXiv  Detail & Related papers  (2023-03-03T11:31:55Z)
• Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data [68.62134204367668]
  This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. The generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution.
  arXiv  Detail & Related papers  (2023-02-14T17:02:35Z)
• How Much is Enough? A Study on Diffusion Times in Score-based Generative Models [76.76860707897413]
  Current best practice advocates for a large T to ensure that the forward dynamics brings the diffusion sufficiently close to a known and simple noise distribution. We show how an auxiliary model can be used to bridge the gap between the ideal and the simulated forward dynamics, followed by a standard reverse diffusion process.
  arXiv  Detail & Related papers  (2022-06-10T15:09:46Z)
• Contrastive Neighborhood Alignment [81.65103777329874]
  We present Contrastive Neighborhood Alignment (CNA), a manifold learning approach to maintain the topology of learned features. The target model aims to mimic the local structure of the source representation space using a contrastive loss. CNA is illustrated in three scenarios: manifold learning, where the model maintains the local topology of the original data in a dimension-reduced space; model distillation, where a small student model is trained to mimic a larger teacher; and legacy model update, where an older model is replaced by a more powerful one.
  arXiv  Detail & Related papers  (2022-01-06T04:58:31Z)
• Scaling Structured Inference with Randomization [64.18063627155128]
  We propose a family of dynamic programming (RDP) randomized for scaling structured models to tens of thousands of latent states. Our method is widely applicable to classical DP-based inference. It is also compatible with automatic differentiation so can be integrated with neural networks seamlessly.
  arXiv  Detail & Related papers  (2021-12-07T11:26:41Z)
• A likelihood approach to nonparametric estimation of a singular distribution using deep generative models [4.329951775163721]
  We investigate a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. We prove that a novel and effective solution exists by perturbing the data with an instance noise. We also characterize the class of distributions that can be efficiently estimated via deep generative models.
  arXiv  Detail & Related papers  (2021-05-09T23:13:58Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.