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Convergence of the denoising diffusion probabilistic models for general noise schedules (2406.01320v5)

Published 3 Jun 2024 in math.PR and stat.ML

Abstract: This work presents a theoretical analysis of the original formulation of denoising diffusion probabilistic models (DDPMs), introduced by Ho, Jain, and Abbeel in Advances in Neural Information Processing Systems, 33 (2020), pp. 6840-6851. An explicit upper bound is derived for the total variation distance between the distribution of the discrete-time DDPM sampling algorithm and a target data distribution, under general noise schedule parameters. The analysis assumes certain technical conditions on the data distribution and a linear growth condition on the noise estimation function. The sampling sequence emerges as an exponential integrator-type approximation of a reverse-time stochastic differential equation (SDE) over a finite time interval. Schr\"odinger's problem provides a tool for estimating the distributional error in reverse time, which connects the reverse-time error with its forward-time counterpart. The score function in DDPMs appears as an adapted solution of a forward-backward SDE, providing a foundation for analyzing the time-discretization error associated with the reverse-time SDE.

Summary

  • The paper rigorously proves that the DDPM sampling algorithm converges weakly to the true data distribution under specific variance and score estimation conditions.
  • It employs a two-step process of forward noise addition and reverse SDE-based recovery, establishing a solid theoretical framework for DDPMs.
  • The analysis informs practical applications in computer vision and medical imaging by detailing convergence dependencies on data dimensions and network design.

Convergence of Denoising Diffusion Probabilistic Models

The paper under discussion offers a thorough theoretical exploration of the convergence characteristics of denoising diffusion probabilistic models (DDPMs). Originally proposed by Ho et al., DDPMs represent a novel class of generative models that have demonstrated significant utility across diverse domains, notably in applications involving computer vision and medical image reconstruction. The central contribution of this work is the rigorous proof of the weak convergence properties of DDPMs, moving from the perspective of practical success to a deeper theoretical understanding.

DDPMs operate on a two-step basis: forward and reverse Markov processes. In the first phase, noise is progressively added to a data distribution until it reaches a Gaussian distribution. Subsequently, in the reverse phase, this process is effectively undone, reconstructing the original data distribution. The paper scrutinizes the original DDPM algorithm, elucidating its convergence using a theoretical framework grounded in stochastic differential equations (SDEs).

Main Theoretical Development

The paper's main theorem asserts the convergence of the distribution generated by the DDPM sampling algorithm to the true data distribution, contingent upon several conditions. These include precise asymptotic behaviors related to parameters such as the variance schedule and score estimation error. The analysis involves representing the sampling sequence via exponential integrator approximations of reverse time SDEs, a strategy that leverages the continuous-time dynamics framework to enhance sampling efficiency.

Conditions and Assumptions

The authors impose boundedness and continuity conditions on the signal-to-noise estimation functions, ensuring controlled approximations to the reverse-time stochastic dynamics. They explore the conditions under which noise variance parameters and score estimation errors diminish appropriately as the number of time steps increases. This aligns with real-world implementation scenarios, as seen in the empirical setups explored by Ho et al.

Key Results

The paper highlights several critical numerical findings. For instance, the convergence rate has dependencies on the dimension of the data and the characteristics of the support set's geometry. The analysis offers insight that is particularly salient for the design of networks used in score estimation. The theoretical innovations presented in the paper present a significant step forward in understanding the stability and representational fidelity of DDPMs.

Implications and Future Directions

The implications of this paper are twofold. Practically, it provides a robust foundation for assessing and enhancing the reliability of DDPMs in generating samples that are statistically similar to the target data distribution. Theoretically, it opens avenues for extending the application of diffusion-based models across other domains by adapting the convergence conditions to different classes of data distributions and network architectures.

The paper also suggests potential future developments, including refining the conditions to accommodate broader classes of distributions or exploring alternative optimization frameworks that retain similar convergence properties while potentially lowering the computational complexity.

In summary, this research presents a sophisticated treatment of the convergence in DDPMs, offering both clarity and rigor to support their continued development and application in machine learning and allied fields.

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