A theory of learning data statistics in diffusion models, from easy to hard

• URL: http://arxiv.org/abs/2603.12901v1
• Date: Fri, 13 Mar 2026 11:07:01 GMT
• Title: A theory of learning data statistics in diffusion models, from easy to hard
• Abstract summary: We show that diffusion models trained on natural images exhibit a distributional simplicity bias, learning simple, pair-wise input statistics before specializing to higher-order correlations.<n>We reproduce this behaviour in simple denoisers trained on a minimal data model, the mixed cumulant model.<n>Our work describes a key mechanism for how diffusion models can learn distributions of increasing complexity.
• Score: 11.841482748098327
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: While diffusion models have emerged as a powerful class of generative models, their learning dynamics remain poorly understood. We address this issue first by empirically showing that standard diffusion models trained on natural images exhibit a distributional simplicity bias, learning simple, pair-wise input statistics before specializing to higher-order correlations. We reproduce this behaviour in simple denoisers trained on a minimal data model, the mixed cumulant model, where we precisely control both pair-wise and higher-order correlations of the inputs. We identify a scalar invariant of the model that governs the sample complexity of learning pair-wise and higher-order correlations that we call the diffusion information exponent, in analogy to related invariants in different learning paradigms. Using this invariant, we prove that the denoiser learns simple, pair-wise statistics of the inputs at linear sample complexity, while more complex higher-order statistics, such as the fourth cumulant, require at least cubic sample complexity. We also prove that the sample complexity of learning the fourth cumulant is linear if pair-wise and higher-order statistics share a correlated latent structure. Our work describes a key mechanism for how diffusion models can learn distributions of increasing complexity.

Related papers

• Multi-Objective Learning for Diffusion Models: A Statistical Theory under Semi-Supervised Learning [83.65494573724904]
  We develop a principled MOL framework for diffusion models with limited data.<n>We propose a two-stage training procedure that first fits lightweight specialist models from limited paired data.<n>We establish bounds generalization showing that the required number of paired samples only depends on the complexity of the specialist model classes.
  arXiv  Detail & Related papers  (2026-05-24T18:19:40Z)
• The Interplay of Data Structure and Imbalance in the Learning Dynamics of Diffusion Models [8.547042933482645]
  We study class-dependent learning in score-based diffusion models.<n>We reveal the explicit hierarchy governing these dynamics.<n>We validate our theoretical predictions empirically using U-Net models trained on Fashion MNIST.
  arXiv  Detail & Related papers  (2026-05-07T14:44:53Z)
• Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity [93.52679474816728]
  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.
  arXiv  Detail & Related papers  (2026-03-21T04:40:19Z)
• Generalization Dynamics of Linear Diffusion Models [8.107431208836426]
  We analytically study the memorisation-to-generalisation transition in a simple model using linear denoisers.<n>Our work clarifies how sample complexity governs generalisation in a simple model of diffusion-based generative models.
  arXiv  Detail & Related papers  (2025-05-30T16:31:58Z)
• Overcoming Dimensional Factorization Limits in Discrete Diffusion Models through Quantum Joint Distribution Learning [79.65014491424151]
  We propose a quantum Discrete Denoising Diffusion Probabilistic Model (QD3PM)<n>It enables joint probability learning through diffusion and denoising in exponentially large Hilbert spaces.<n>This paper establishes a new theoretical paradigm in generative models by leveraging the quantum advantage in joint distribution learning.
  arXiv  Detail & Related papers  (2025-05-08T11:48:21Z)
• Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure [8.320632531909682]
  We study the generalizability of diffusion models by looking into the hidden properties of the learned score functions.<n>As diffusion models transition from memorization to generalization, their corresponding nonlinear diffusion denoisers exhibit increasing linearity.
  arXiv  Detail & Related papers  (2024-10-31T15:57:04Z)
• Provable Statistical Rates for Consistency Diffusion Models [87.28777947976573]
  Despite the state-of-the-art performance, diffusion models are known for their slow sample generation due to the extensive number of steps involved. This paper contributes towards the first statistical theory for consistency models, formulating their training as a distribution discrepancy minimization problem.
  arXiv  Detail & Related papers  (2024-06-23T20:34:18Z)
• Ablation Based Counterfactuals [7.481286710933861]
  Ablation Based Counterfactuals (ABC) is a method of performing counterfactual analysis that relies on model ablation rather than model retraining. We demonstrate how we can construct a model like this using an ensemble of diffusion models. We then use this model to study the limits of training data attribution by enumerating full counterfactual landscapes.
  arXiv  Detail & Related papers  (2024-06-12T06:22:51Z)
• DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion [66.21290235237808]
  We introduce an energy constrained diffusion model which encodes a batch of instances from a dataset into evolutionary states. We provide rigorous theory that implies closed-form optimal estimates for the pairwise diffusion strength among arbitrary instance pairs. Experiments highlight the wide applicability of our model as a general-purpose encoder backbone with superior performance in various tasks.
  arXiv  Detail & Related papers  (2023-01-23T15:18:54Z)
• Good Classifiers are Abundant in the Interpolating Regime [64.72044662855612]
  We develop a methodology to compute precisely the full distribution of test errors among interpolating classifiers. We find that test errors tend to concentrate around a small typical value $varepsilon*$, which deviates substantially from the test error of worst-case interpolating model. Our results show that the usual style of analysis in statistical learning theory may not be fine-grained enough to capture the good generalization performance observed in practice.
  arXiv  Detail & Related papers  (2020-06-22T21:12:31Z)

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.