Manifold Generalization Provably Proceeds Memorization in Diffusion Models

• URL: http://arxiv.org/abs/2603.23792v1
• Date: Tue, 24 Mar 2026 23:50:09 GMT
• Title: Manifold Generalization Provably Proceeds Memorization in Diffusion Models
• Abstract summary: Diffusion models often generate novel samples even when the learned score is only emphcoarse.<n>We prove that diffusion models trained with coarse scores can exploit the emphregularity of the manifold support.
• Score: 33.15269246693525
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Diffusion models often generate novel samples even when the learned score is only \emph{coarse} -- a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the \emph{manifold hypothesis}, this behavior can instead be explained by coarse scores capturing the \emph{geometry} of the data while discarding the fine-scale distributional structure of the population measure~$μ_{\scriptscriptstyle\mathrm{data}}$. Concretely, whereas estimating the full data distribution $μ_{\scriptscriptstyle\mathrm{data}}$ supported on a $k$-dimensional manifold is known to require the classical minimax rate $\tilde{\mathcal{O}}(N^{-1/k})$, we prove that diffusion models trained with coarse scores can exploit the \emph{regularity of the manifold support} and attain a near-parametric rate toward a \emph{different} target distribution. This target distribution has density uniformly comparable to that of~$μ_{\scriptscriptstyle\mathrm{data}}$ throughout any $\tilde{\mathcal{O}}\bigl(N^{-β/(4k)}\bigr)$-neighborhood of the manifold, where $β$ denotes the manifold regularity. Our guarantees therefore depend only on the smoothness of the underlying support, and are especially favorable when the data density itself is irregular, for instance non-differentiable. In particular, when the manifold is sufficiently smooth, we obtain that \emph{generalization} -- formalized as the ability to generate novel, high-fidelity samples -- occurs at a statistical rate strictly faster than that required to estimate the full population distribution~$μ_{\scriptscriptstyle\mathrm{data}}$.

Related papers

• Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data [32.72306410557258]
  We study the statistical convergence of score-based diffusion models for learning an unknown distribution $$ from finitely many samples.<n>Our results demonstrate that diffusion models naturally adapt to the intrinsic geometry of data.<n>Our theory conceptually bridges the analysis of diffusion models with that of GANs and the sharp minimax rates established in optimal transport.
  arXiv  Detail & Related papers  (2026-03-04T03:59:02Z)
• Optimal Convergence Analysis of DDPM for General Distributions [11.155024379105788]
  Denoising Diffusion Probabilistic Model (DDPM) is one of the most widely used samplers.<n>We provide a refined convergence analysis of the DDPM sampler.<n>We show that our convergence analysis is tight for a wide array of target distributions.
  arXiv  Detail & Related papers  (2025-10-31T15:44:50Z)
• Computational bottlenecks for denoising diffusions [8.05574597775852]
  We show that there exist drifts that are superpolynomially close to the optimum value and yet yield samples with distribution that is very far from the target one.<n>We provide evidence to the contrary by studying a probability distribution $mu$ for which sampling is easy, but the drift of the diffusion process is intractable.
  arXiv  Detail & Related papers  (2025-03-11T04:21:01Z)
• Outsourced diffusion sampling: Efficient posterior inference in latent spaces of generative models [65.71506381302815]
  We propose amortize the cost of sampling from a posterior distribution of the form $p(mathbfxmidmathbfy) propto p_theta(mathbfx)$.<n>For many models and constraints, the posterior in noise space is smoother than in data space, making it more suitable for amortized inference.
  arXiv  Detail & Related papers  (2025-02-10T19:49:54Z)
• Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers [49.97755400231656]
  We present the first performance guarantee with explicit dimensional dependencies for general score-mismatched diffusion samplers.<n>We show that score mismatches result in an distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions.<n>This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise.
  arXiv  Detail & Related papers  (2024-10-17T16:42:12Z)
• Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat [49.1574468325115]
  This paper presents explicit non-asymptotic bounds on the forward diffusion error in total variation (TV)<n>We parametrise multi-modal data distributions in terms of the distance $R$ to their furthest modes and consider forward diffusions with additive and multiplicative noise.
  arXiv  Detail & Related papers  (2024-08-25T10:28:31Z)
• Rejection via Learning Density Ratios [50.91522897152437]
  Classification with rejection emerges as a learning paradigm which allows models to abstain from making predictions.<n>We propose a different distributional perspective, where we seek to find an idealized data distribution which maximizes a pretrained model's performance.<n>Our framework is tested empirically over clean and noisy datasets.
  arXiv  Detail & Related papers  (2024-05-29T01:32:17Z)
• Convergence Analysis of Probability Flow ODE for Score-based Generative Models [5.939858158928473]
  We study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives.<n>We prove the total variation between the target and the generated data distributions can be bounded above by $mathcalO(d3/4delta1/2)$ in the continuous time level.
  arXiv  Detail & Related papers  (2024-04-15T12:29:28Z)
• Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions [11.222970035173372]
  kernel-based score estimator achieves an optimal mean square error of $widetildeOleft(n-1 t-fracd+22(tfracd2 vee 1)right) We show that a kernel-based score estimator achieves an optimal mean square error of $widetildeOleft(n-1/2 t-fracd4right)$ upper bound for the total variation error of the distribution of the sample generated by the diffusion model under a mere sub-Gaussian
  arXiv  Detail & Related papers  (2024-02-23T20:51:31Z)
• Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach [49.97755400231656]
  We show that a new accelerated DDPM sampler achieves accelerated performance for three broad distribution classes not considered before.<n>Our results show an improved dependency on the data dimension $d$ among accelerated DDPM type samplers.
  arXiv  Detail & Related papers  (2024-02-21T16:11:47Z)
• Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models [49.81937966106691]
  We develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach.
  arXiv  Detail & Related papers  (2023-06-15T16:30:08Z)
• Beyond Black Box Densities: Parameter Learning for the Deviated Components [15.501680326749515]
  A known density function estimate may have been previously obtained by a black box method. The increased complexity of the data set may result in the true density being deviated from the known estimate by a mixture distribution. We establish rates of convergence for the maximum likelihood estimates of $lambda*$ and $G*$ under Wasserstein metric.
  arXiv  Detail & Related papers  (2022-02-05T22:44:20Z)

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.