Convergence Analysis of Probability Flow ODE for Score-based Generative Models

• URL: http://arxiv.org/abs/2404.09730v2
• Date: Sun, 21 Jul 2024 20:23:07 GMT
• Title: Convergence Analysis of Probability Flow ODE for Score-based Generative Models
• Authors: Daniel Zhengyu Huang, Jiaoyang Huang, Zhengjiang Lin,
• Abstract summary: We study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. 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.
• Score: 5.939858158928473
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. Assuming access to $L^2$-accurate estimates of the score function, we prove the total variation between the target and the generated data distributions can be bounded above by $\mathcal{O}(d^{3/4}\delta^{1/2})$ in the continuous time level, where $d$ denotes the data dimension and $\delta$ represents the $L^2$-score matching error. For practical implementations using a $p$-th order Runge-Kutta integrator with step size $h$, we establish error bounds of $\mathcal{O}(d^{3/4}\delta^{1/2} + d\cdot(dh)^p)$ at the discrete level. Finally, we present numerical studies on problems up to 128 dimensions to verify our theory.

Related papers

• Entangled Mean Estimation in High-Dimensions [36.97113089188035]
  We study the task of high-dimensional entangled mean estimation in the subset-of-signals model. We show that the optimal error (up to polylogarithmic factors) is $f(alpha,N) + sqrtD/(alpha N)$, where the term $f(alpha,N)$ is the error of the one-dimensional problem and the second term is the sub-Gaussian error rate.
  arXiv  Detail & Related papers  (2025-01-09T18:31:35Z)
• O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions [6.76974373198208]
  We establish a fast convergence theory for the denoising diffusion probabilistic model (DDPM) under minimal assumptions. We show that the convergence rate improves to $O(k/T)$, where $k$ is the intrinsic dimension of the target data distribution. This highlights the ability of DDPM to automatically adapt to unknown low-dimensional structures.
  arXiv  Detail & Related papers  (2024-09-27T17:59:10Z)
• A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models [45.60426164657739]
  We develop non-asymptotic convergence theory for a diffusion-based sampler. We prove that $d/varepsilon$ are sufficient to approximate the target distribution to within $varepsilon$ total-variation distance. Our results also characterize how $ell$ score estimation errors affect the quality of the data generation processes.
  arXiv  Detail & Related papers  (2024-08-05T09:02:24Z)
• Effective Minkowski Dimension of Deep Nonparametric Regression: Function Approximation and Statistical Theories [70.90012822736988]
  Existing theories on deep nonparametric regression have shown that when the input data lie on a low-dimensional manifold, deep neural networks can adapt to intrinsic data structures. This paper introduces a relaxed assumption that input data are concentrated around a subset of $mathbbRd$ denoted by $mathcalS$, and the intrinsic dimension $mathcalS$ can be characterized by a new complexity notation -- effective Minkowski dimension.
  arXiv  Detail & Related papers  (2023-06-26T17:13:31Z)
• 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)
• Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models [91.22420505636006]
  This paper deals with the problem of efficient sampling from a differential equation, given the drift function and the diffusion matrix. It is possible to obtain independent and identically distributed (i.i.d.) samples at precision $varepsilon$ with a cost that is $m2 d log (1/varepsilon)$ Our results suggest that as the true solution gets smoother, we can circumvent the curse of dimensionality without requiring any sort of convexity.
  arXiv  Detail & Related papers  (2023-03-30T02:50:49Z)
• Uncertainty Quantification of MLE for Entity Ranking with Covariates [3.2839905453386162]
  This paper concerns with statistical estimation and inference for the ranking problems based on pairwise comparisons. We propose a novel model, Co-Assisted Ranking Estimation (CARE) model, that extends the well-known Bradley-Terry-Luce (BTL) model. We derive the maximum likelihood estimator of $alpha_i*_i=1n$ and $beta*$ under a sparse comparison graph. We validate our theoretical results through large-scale numerical studies and an application to the mutual fund stock holding dataset.
  arXiv  Detail & Related papers  (2022-12-20T02:28:27Z)
• Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions [9.953088581242845]
  We provide convergence guarantees with complexity for any data distribution with second-order moment. Our result does not rely on any log-concavity or functional inequality assumption. Our theoretical analysis provides comparison between different discrete approximations and may guide the choice of discretization points in practice.
  arXiv  Detail & Related papers  (2022-11-03T15:51:00Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)

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.