# Convergence for score-based generative modeling with polynomial complexity

@article{Lee2022ConvergenceFS, title={Convergence for score-based generative modeling with polynomial complexity}, author={Holden Lee and Jianfeng Lu and Yixin Tan}, journal={ArXiv}, year={2022}, volume={abs/2206.06227} }

Score-based generative modeling (SGM) is a highly successful approach for learning a probability distribution from data and generating further samples. We prove the first polynomial convergence guarantees for the core mechanic behind SGM: drawing samples from a probability density p given a score estimate (an estimate of ∇ ln p) that is accurate in L(p). Compared to previous works, we do not incur error that grows exponentially in time or that suffers from a curse of dimensionality. Our…

## 16 Citations

### Convergence of score-based generative modeling for general data distributions

- Computer Science, MathematicsALT
- 2023

This work considers a popular kind of SGM -- denoising diffusion models -- and gives polynomial convergence guarantees for general data distributions, with no assumptions related to functional inequalities or smoothness.

### Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions

- Computer ScienceArXiv
- 2022

It is shown that score-based generative models such as denoising diﬀusion probabilistic models (DDPMs) can eﬃciently sample from essentially any realistic data distribution, and theoretical convergence guarantees for these models hold for an L 2 -accurate score estimate.

### Convergence in KL Divergence of the Inexact Langevin Algorithm with Application to Score-based Generative Models

- Computer Science, MathematicsArXiv
- 2022

The Inexact Langevin Algorithm for sampling using estimated score function when the target distribution satisﬁes log-Sobolev inequality (LSI) is studied, motivated by Score-based Generative Modeling (SGM), and a long-term convergence in Kullback-Leibler divergence is proved.

### Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions

- Computer Science, MathematicsArXiv
- 2022

Under an L 2 -accurate score estimator, convergence guarantees with polynomial complexity for any data distribution with second-order moment are provided, by either employing an early stopping technique or assuming smoothness condition on the score function of the data distribution.

### Statistical Efficiency of Score Matching: The View from Isoperimetry

- Computer Science, MathematicsArXiv
- 2022

This paper shows that the score matching estimator is statistically comparable to the maximum likelihood when the distribution has a small isoperimetric constant, and shows a direct parallel in the discrete setting, where it connects the statistical properties of pseudolikelihood estimation with approximate tensorization of entropy and the Glauber dynamics.

### Proposal of a Score Based Approach to Sampling Using Monte Carlo Estimation of Score and Oracle Access to Target Density

- Computer ScienceArXiv
- 2022

This work considers if the authors have no initial samples from the target density, but rather 0 th and 1 st order oracle access to the log likelihood, and proposes a Monte Carlo method to estimate the score empirically as a particular expectation of a random variable.

### Convergence of denoising diffusion models under the manifold hypothesis

- Mathematics, Computer ScienceArXiv
- 2022

This paper provides the first convergence results for diffusion models in this setting by providing quantitative bounds on the Wasserstein distance of order one between the target data distribution and the generative distribution of the diffusion model.

### How to Trust Your Diffusion Model: A Convex Optimization Approach to Conformal Risk Control

- Computer ScienceArXiv
- 2023

This work focuses on image-to-image regression tasks and presents a generalization of the Risk-Controlling Prediction Sets procedure, that allows to provide entrywise calibrated intervals for future samples of any diffusion model, and control a certain notion of risk with respect to a ground truth image with minimal mean interval length.

### Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers

- MathematicsArXiv
- 2023

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's…

### Boundary Guided Mixing Trajectory for Semantic Control with Diffusion Models

- Computer ScienceArXiv
- 2023

This work achieves SOTA semantic control performance on various application settings by optimizing the denoising trajectory solely via frozen DDMs through a more comprehensive understanding of the intermediate high-dimensional latent spaces by theoretically and empirically analyzing their probabilistic and geometric behaviors in the Markov chain.

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