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Critically-Damped Higher-Order Langevin Dynamics for Generative Modeling

Benjamin Sterling, Chad Gueli, Mónica F. Bugallo

TL;DR

The paper tackles slow convergence in higher-order diffusion models by introducing critically damped higher-order Langevin dynamics (HOLD++). It derives closed-form forward process mean and covariance and constructs an $n$-th order forward SDE with matrices $\mathbf{F}$ and $\mathbf{G}$ designed to yield a single geometric eigenvalue under critical damping. It proves that, for a fixed forward-trace, the critically damped parameter set minimizes the maximum real part of the eigenvalues, establishing optimality. Empirically, HOLD++ is validated on CIFAR-10 and CelebA-HQ $256\times256$, showing improved convergence and competitive FID scores, with $n=3$ often performing best on CIFAR-10.

Abstract

Denoising diffusion probabilistic models (DDPMs) represent an entirely new class of generative AI methods that have yet to be fully explored. They use Langevin dynamics, represented as stochastic differential equations, to describe a process that transforms data into noise, the forward process, and a process that transforms noise into generated data, the reverse process. Many of these methods utilize auxiliary variables that formulate the data as a ``position" variable, and the auxiliary variables are referred to as ``velocity", ``acceleration", etc. In this sense, it is possible to ``critically damp" the dynamics. Critical damping has been successfully introduced in Critically-Damped Langevin Dynamics (CLD) and Critically-Damped Third-Order Langevin Dynamics (TOLD++), but has not yet been applied to dynamics of arbitrary order. The proposed methodology generalizes Higher-Order Langevin Dynamics (HOLD), a recent state-of-the-art diffusion method, by introducing the concept of critical damping from systems analysis. Similarly to TOLD++, this work proposes an optimal set of hyperparameters in the $n$-dimensional case, where HOLD leaves these to be user defined. Additionally, this work provides closed-form solutions for the mean and covariance of the forward process that greatly simplify its implementation. Experiments are performed on the CIFAR-10 and CelebA-HQ $256 \times 256$ datasets, and validated against the FID metric.

Critically-Damped Higher-Order Langevin Dynamics for Generative Modeling

TL;DR

The paper tackles slow convergence in higher-order diffusion models by introducing critically damped higher-order Langevin dynamics (HOLD++). It derives closed-form forward process mean and covariance and constructs an -th order forward SDE with matrices and designed to yield a single geometric eigenvalue under critical damping. It proves that, for a fixed forward-trace, the critically damped parameter set minimizes the maximum real part of the eigenvalues, establishing optimality. Empirically, HOLD++ is validated on CIFAR-10 and CelebA-HQ , showing improved convergence and competitive FID scores, with often performing best on CIFAR-10.

Abstract

Denoising diffusion probabilistic models (DDPMs) represent an entirely new class of generative AI methods that have yet to be fully explored. They use Langevin dynamics, represented as stochastic differential equations, to describe a process that transforms data into noise, the forward process, and a process that transforms noise into generated data, the reverse process. Many of these methods utilize auxiliary variables that formulate the data as a ``position" variable, and the auxiliary variables are referred to as ``velocity", ``acceleration", etc. In this sense, it is possible to ``critically damp" the dynamics. Critical damping has been successfully introduced in Critically-Damped Langevin Dynamics (CLD) and Critically-Damped Third-Order Langevin Dynamics (TOLD++), but has not yet been applied to dynamics of arbitrary order. The proposed methodology generalizes Higher-Order Langevin Dynamics (HOLD), a recent state-of-the-art diffusion method, by introducing the concept of critical damping from systems analysis. Similarly to TOLD++, this work proposes an optimal set of hyperparameters in the -dimensional case, where HOLD leaves these to be user defined. Additionally, this work provides closed-form solutions for the mean and covariance of the forward process that greatly simplify its implementation. Experiments are performed on the CIFAR-10 and CelebA-HQ datasets, and validated against the FID metric.

Paper Structure

This paper contains 16 sections, 9 theorems, 80 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. The Simple Case, $n=1$
  4. Methods
  5. Optimality of Critical Damping
  6. Critical Damping
  7. From Matrices to Polynomials
  8. From Polynomials to Their Coefficients
  9. Experiments
  10. CIFAR-10
  11. CelebA-HQ $256\times 256$
  12. Conclusion
  13. Proof of Lemma \ref{['lem:dj']}
  14. Proof of Theorem \ref{['thm:aj']}
  15. Proof of Theorem \ref{['thm:stog']}
  16. ...and 1 more sections

Key Result

Theorem 1

The solutions to differential equations eq:mean_eq and eq:var_eq are

Figures (2)

  • Figure 1: CIFAR-10 images generated at $450{,}000$ training iterations for model orders 2 and 3.
  • Figure 2: Generated samples from CelebA-HQ for model order 3 without cherry picking

Theorems & Definitions (17)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 2
  • Theorem 4
  • Theorem 5
  • ...and 7 more