Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity

• URL: http://arxiv.org/abs/2603.20645v1
• Date: Sat, 21 Mar 2026 04:40:19 GMT
• Title: Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
• Abstract summary: Diffusion models have become a leading framework in generative modeling.<n>This paper investigates how diffusion models learn structured data, focusing on two key aspects: statistical complexity and influence of data geometric properties.<n>We also highlight the interplay of manifold curvature with the structures in the score function.
• Score: 93.52679474816728
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Diffusion models have become a leading framework in generative modeling, yet their theoretical understanding -- especially for high-dimensional data concentrated on low-dimensional structures -- remains incomplete. This paper investigates how diffusion models learn such structured data, focusing on two key aspects: statistical complexity and influence of data geometric properties. By modeling data as samples from a smooth Riemannian manifold, our analysis reveals crucial decompositions of score functions in diffusion models under different levels of injected noise. We also highlight the interplay of manifold curvature with the structures in the score function. These analyses enable an efficient neural network approximation to the score function, built upon which we further provide statistical rates for score estimation and distribution learning. Remarkably, the obtained statistical rates are governed by the intrinsic dimension of data and the manifold curvature. These results advance the statistical foundations of diffusion models, bridging theory and practice for generative modeling on manifolds.

Related papers

• Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Data [7.047721358838963]
  We study the theoretical behavior of denoising score matching.<n>We derive exact expressions for the test, train, and score errors in the high-dimensional limit.
  arXiv  Detail & Related papers  (2026-03-24T08:58:12Z)
• Nonparametric Data Attribution for Diffusion Models [57.820618036556084]
  Data attribution for generative models seeks to quantify the influence of individual training examples on model outputs.<n>We propose a nonparametric attribution method that operates entirely on data, measuring influence via patch-level similarity between generated and training images.
  arXiv  Detail & Related papers  (2025-10-16T03:37:16Z)
• Improved Sample Complexity For Diffusion Model Training Without Empirical Risk Minimizer Access [47.96419637803502]
  We present a principled theoretical framework analyzing diffusion models, providing a state-of-the-art sample complexity bound to $widetildemathcalO(epsilon-4)$.<n>Our structured decomposition of the score estimation error into statistical and optimization components offers critical insights into how diffusion models can be trained efficiently.
  arXiv  Detail & Related papers  (2025-05-23T20:02:15Z)
• Data augmentation using diffusion models to enhance inverse Ising inference [2.654300333196867]
  We show that diffusion models can enhance parameter inference by augmenting small datasets.<n>This study serves as a proof-of-concept for using diffusion models for data augmentation in physics-related problems.
  arXiv  Detail & Related papers  (2025-03-13T08:29:17Z)
• Continuous Diffusion Model for Language Modeling [64.7425225935854]
  Existing continuous diffusion models for discrete data underperform compared to discrete methods.<n>We propose a continuous diffusion model for language modeling that incorporates the geometry of the underlying categorical distribution.<n>Our method outperforms existing discrete diffusion models and approaches the performance of autoregressive models.
  arXiv  Detail & Related papers  (2025-02-17T08:54:29Z)
• Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory [87.00653989457834]
  Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. Despite the empirical success, theory of conditional diffusion models is largely missing. This paper bridges the gap by presenting a sharp statistical theory of distribution estimation using conditional diffusion models.
  arXiv  Detail & Related papers  (2024-03-18T17:08:24Z)
• Dynamical Regimes of Diffusion Models [14.797301819675454]
  We study generative diffusion models in the regime where the dimension of space and the number of data are large. Our analysis reveals three distinct dynamical regimes during the backward generative diffusion process. The dependence of the collapse time on the dimension and number of data provides a thorough characterization of the curse of dimensionality for diffusion models.
  arXiv  Detail & Related papers  (2024-02-28T17:19:26Z)
• Towards Theoretical Understandings of Self-Consuming Generative Models [56.84592466204185]
  This paper tackles the emerging challenge of training generative models within a self-consuming loop. We construct a theoretical framework to rigorously evaluate how this training procedure impacts the data distributions learned by future models. We present results for kernel density estimation, delivering nuanced insights such as the impact of mixed data training on error propagation.
  arXiv  Detail & Related papers  (2024-02-19T02:08:09Z)
• 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)
• 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)

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.