Polynomial Convergence of Riemannian Diffusion Models

• URL: http://arxiv.org/abs/2601.02499v1
• Date: Mon, 05 Jan 2026 19:14:09 GMT
• Title: Polynomial Convergence of Riemannian Diffusion Models
• Authors: Xingyu Xu, Ziyi Zhang, Yorie Nakahira, Guannan Qu, Yuejie Chi,
• Abstract summary: Diffusion models are considered one of the state-of-the-art generative models in modern AI.<n>Most of the existing literature assumes that the underlying space is Euclidean.<n>In many practical applications, the data are constrained to lie on a submanifold of Euclidean space.
• Score: 46.72936436234762
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Diffusion models have demonstrated remarkable empirical success in the recent years and are considered one of the state-of-the-art generative models in modern AI. These models consist of a forward process, which gradually diffuses the data distribution to a noise distribution spanning the whole space, and a backward process, which inverts this transformation to recover the data distribution from noise. Most of the existing literature assumes that the underlying space is Euclidean. However, in many practical applications, the data are constrained to lie on a submanifold of Euclidean space. Addressing this setting, De Bortoli et al. (2022) introduced Riemannian diffusion models and proved that using an exponentially small step size yields a small sampling error in the Wasserstein distance, provided the data distribution is smooth and strictly positive, and the score estimate is $L_\infty$-accurate. In this paper, we greatly strengthen this theory by establishing that, under $L_2$-accurate score estimate, a {\em polynomially small stepsize} suffices to guarantee small sampling error in the total variation distance, without requiring smoothness or positivity of the data distribution. Our analysis only requires mild and standard curvature assumptions on the underlying manifold. The main ingredients in our analysis are Li-Yau estimate for the log-gradient of heat kernel, and Minakshisundaram-Pleijel parametrix expansion of the perturbed heat equation. Our approach opens the door to a sharper analysis of diffusion models on non-Euclidean spaces.

Related papers

• High-accuracy and dimension-free sampling with diffusions [27.7060066305274]
  We propose a new solver for diffusion models relying on a subtle interplay between low-degree approximation and the collocation method.<n>We prove that its complexity scales emphpolylogarithmically in $1/varepsilon$, yielding the first high-accuracy' guarantee for a diffusion-based sampler.
  arXiv  Detail & Related papers  (2026-01-15T18:58:50Z)
• Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces [5.716752583983991]
  When the data distribution consists of n points, empirical diffusion models tend to reproduce existing data points.<n>This work shows that the memorization issue can be solved simply by applying an inertia update at the end of the empirical diffusion simulation.<n>We demonstrate that the distribution of samples from this model approximates the true data distribution on a $C2$ manifold of dimension $d$, within a Wasserstein-1 distance of order $O(n-frac2d+4)$.
  arXiv  Detail & Related papers  (2025-05-05T09:40:41Z)
• On the Wasserstein Convergence and Straightness of Rectified Flow [54.580605276017096]
  Rectified Flow (RF) is a generative model that aims to learn straight flow trajectories from noise to data.<n>We provide a theoretical analysis of the Wasserstein distance between the sampling distribution of RF and the target distribution.<n>We present general conditions guaranteeing uniqueness and straightness of 1-RF, which is in line with previous empirical findings.
  arXiv  Detail & Related papers  (2024-10-19T02:36:11Z)
• Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions [6.9408143976091745]
  Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions.<n>We study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of score learning.<n>In terms of sampling complexity, we obtain rates independent of the ambient dimension w.r.t. the Kullback-Leibler divergence, and $O(sqrtD)$ w.r.t. the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-09-27T14:57:18Z)
• Blessing of Dimensionality for Approximating Sobolev Classes on Manifolds [14.183849746284816]
  We consider optimal uniform approximations with functions of finite statistical complexity.<n>In particular, we demonstrate that the statistical complexity required to approximate a class of bounded Sobolev functions on a compact manifold is bounded from below.
  arXiv  Detail & Related papers  (2024-08-13T15:56:42Z)
• Bayesian Circular Regression with von Mises Quasi-Processes [57.88921637944379]
  In this work we explore a family of expressive and interpretable distributions over circle-valued random functions.<n>For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Gibbs sampling.<n>We present experiments applying this model to the prediction of wind directions and the percentage of the running gait cycle as a function of joint angles.
  arXiv  Detail & Related papers  (2024-06-19T01:57:21Z)
• Scaling Riemannian Diffusion Models [68.52820280448991]
  We show that our method enables us to scale to high dimensional tasks on nontrivial manifold. We model QCD densities on $SU(n)$ lattices and contrastively learned embeddings on high dimensional hyperspheres.
  arXiv  Detail & Related papers  (2023-10-30T21:27:53Z)
• Conformal inference for regression on Riemannian Manifolds [45.560812800359685]
  We investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by $X$, lies in an Euclidean space.<n>We prove the almost sure convergence of the empirical version of these regions on the manifold to their population counterparts.
  arXiv  Detail & Related papers  (2023-10-12T10:56:25Z)
• Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes [57.396578974401734]
  We introduce a principled framework for building a generative diffusion process on general manifold. Instead of following the denoising approach of previous diffusion models, we construct a diffusion process using a mixture of bridge processes. We develop a geometric understanding of the mixture process, deriving the drift as a weighted mean of tangent directions to the data points.
  arXiv  Detail & Related papers  (2023-10-11T06:04:40Z)
• Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution [2.1146241717926664]
  We show that the Wasserstein GAN, constrained to left-invertible push-forward maps, generates distributions that avoid replication and significantly deviate from the empirical distribution. Our most important contribution provides a finite-sample lower bound on the Wasserstein-1 distance between the generative distribution and the empirical one. We also establish a finite-sample upper bound on the distance between the generative distribution and the true data-generating one.
  arXiv  Detail & Related papers  (2023-07-31T06:11:57Z)
• 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)

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.