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Critically Damped Third-Order Langevin Dynamics

Benjamin Sterling, Mónica F. Bugallo

TL;DR

This work introduces TOLD++, a critical-damping enhancement for Third-Order Langevin Dynamics in diffusion models. By performing eigenanalysis on the forward transition matrix, it selects $(\gamma,\xi)=(2\sqrt{2},3\sqrt{3})$ to induce a single, rapidly decaying eigenvalue $-\sqrt{3}$, achieving faster convergence without extra cost. The authors provide analytic expressions for the forward distribution and show asymptotic agreement with the original TOLD, while demonstrating superior empirical performance on synthetic data (Gaussian mixtures and Swiss Roll) and CIFAR-10 via lower FID scores. The approach offers a principled, theoretically grounded improvement to diffusion-based sampling with potential applicability to higher-order Langevin dynamics.

Abstract

While systems analysis has been studied for decades in the context of control theory, it has only been recently used to improve the convergence of Denoising Diffusion Probabilistic Models. This work describes a novel improvement to Third- Order Langevin Dynamics (TOLD), a recent diffusion method that performs better than its predecessors. This improvement, abbreviated TOLD++, is carried out by critically damping the TOLD forward transition matrix similarly to Dockhorn's Critically-Damped Langevin Dynamics (CLD). Specifically, it exploits eigen-analysis of the forward transition matrix to derive the optimal set of dynamics under the original TOLD scheme. TOLD++ is theoretically guaranteed to converge faster than TOLD, and its faster convergence is verified on the Swiss Roll toy dataset and CIFAR-10 dataset according to the FID metric.

Critically Damped Third-Order Langevin Dynamics

TL;DR

This work introduces TOLD++, a critical-damping enhancement for Third-Order Langevin Dynamics in diffusion models. By performing eigenanalysis on the forward transition matrix, it selects to induce a single, rapidly decaying eigenvalue , achieving faster convergence without extra cost. The authors provide analytic expressions for the forward distribution and show asymptotic agreement with the original TOLD, while demonstrating superior empirical performance on synthetic data (Gaussian mixtures and Swiss Roll) and CIFAR-10 via lower FID scores. The approach offers a principled, theoretically grounded improvement to diffusion-based sampling with potential applicability to higher-order Langevin dynamics.

Abstract

While systems analysis has been studied for decades in the context of control theory, it has only been recently used to improve the convergence of Denoising Diffusion Probabilistic Models. This work describes a novel improvement to Third- Order Langevin Dynamics (TOLD), a recent diffusion method that performs better than its predecessors. This improvement, abbreviated TOLD++, is carried out by critically damping the TOLD forward transition matrix similarly to Dockhorn's Critically-Damped Langevin Dynamics (CLD). Specifically, it exploits eigen-analysis of the forward transition matrix to derive the optimal set of dynamics under the original TOLD scheme. TOLD++ is theoretically guaranteed to converge faster than TOLD, and its faster convergence is verified on the Swiss Roll toy dataset and CIFAR-10 dataset according to the FID metric.
Paper Structure (7 sections, 1 theorem, 23 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 7 sections, 1 theorem, 23 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Methodology
  4. Experiments and Results
  5. Swiss Roll dataset
  6. CIFAR-10 dataset
  7. Conclusion

Key Result

Lemma 3.1

Suppose $\lambda_1, \lambda_2, \lambda_3 \in \mathbb{R}$ satisfy $p_F(\lambda) = 0$ and $\lambda_1 \leq \lambda_2 \leq \lambda_3$. It follows that $\min(\lambda_3) = -\sqrt{3}.$

Figures (2)

  • Figure 1: Experiment on TOLD vs TOLD++ densities on samples from a Gaussian Mixture Model. The lighter the color, the more intense the density. The vertical dashed line occurs at the same diffusion time, and displays that TOLD++ converges faster.
  • Figure 2: Generated CIFAR-10 samples at 790,000 training iterations, without cherry picking.

Theorems & Definitions (2)

  • Lemma 3.1
  • proof