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Tackling the Singularities at the Endpoints of Time Intervals in Diffusion Models

Pengze Zhang, Hubery Yin, Chen Li, Xiaohua Xie

TL;DR

This work tackles the endpoint singularities in diffusion models, showing that the reverse process can be treated as Gaussian across time and that the singularities at $t=0$ are intrinsic while those at $t=1$ are removable. Building on this theory, it introduces SingDiffusion, a plug-and-play module that adjusts the initial sampling step to correct the average brightness issue without retraining. Empirical results demonstrate substantial improvements in brightness control and image fidelity (lower FID, higher CLIP alignment) across multiple pre-trained models and even when integrated with ControlNet. The method is trained once on a large image-text corpus and is broadly applicable, enhancing generation quality with minimal training burden.

Abstract

Most diffusion models assume that the reverse process adheres to a Gaussian distribution. However, this approximation has not been rigorously validated, especially at singularities, where t=0 and t=1. Improperly dealing with such singularities leads to an average brightness issue in applications, and limits the generation of images with extreme brightness or darkness. We primarily focus on tackling singularities from both theoretical and practical perspectives. Initially, we establish the error bounds for the reverse process approximation, and showcase its Gaussian characteristics at singularity time steps. Based on this theoretical insight, we confirm the singularity at t=1 is conditionally removable while it at t=0 is an inherent property. Upon these significant conclusions, we propose a novel plug-and-play method SingDiffusion to address the initial singular time step sampling, which not only effectively resolves the average brightness issue for a wide range of diffusion models without extra training efforts, but also enhances their generation capability in achieving notable lower FID scores.

Tackling the Singularities at the Endpoints of Time Intervals in Diffusion Models

TL;DR

This work tackles the endpoint singularities in diffusion models, showing that the reverse process can be treated as Gaussian across time and that the singularities at are intrinsic while those at are removable. Building on this theory, it introduces SingDiffusion, a plug-and-play module that adjusts the initial sampling step to correct the average brightness issue without retraining. Empirical results demonstrate substantial improvements in brightness control and image fidelity (lower FID, higher CLIP alignment) across multiple pre-trained models and even when integrated with ControlNet. The method is trained once on a large image-text corpus and is broadly applicable, enhancing generation quality with minimal training burden.

Abstract

Most diffusion models assume that the reverse process adheres to a Gaussian distribution. However, this approximation has not been rigorously validated, especially at singularities, where t=0 and t=1. Improperly dealing with such singularities leads to an average brightness issue in applications, and limits the generation of images with extreme brightness or darkness. We primarily focus on tackling singularities from both theoretical and practical perspectives. Initially, we establish the error bounds for the reverse process approximation, and showcase its Gaussian characteristics at singularity time steps. Based on this theoretical insight, we confirm the singularity at t=1 is conditionally removable while it at t=0 is an inherent property. Upon these significant conclusions, we propose a novel plug-and-play method SingDiffusion to address the initial singular time step sampling, which not only effectively resolves the average brightness issue for a wide range of diffusion models without extra training efforts, but also enhances their generation capability in achieving notable lower FID scores.
Paper Structure (42 sections, 16 theorems, 70 equations, 10 figures, 4 tables, 2 algorithms)

This paper contains 42 sections, 16 theorems, 70 equations, 10 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Reverse process approximation
  4. Singularities in diffusion model
  5. Average brightness issue
  6. Method
  7. Preliminaries
  8. Error bound estimation
  9. Tackling the singularities
  10. The singularities at $t=1$
  11. The singularities at $t=0$
  12. SingDiffusion
  13. Experiment
  14. Implement details
  15. Average brightness issue
  16. ...and 27 more sections

Key Result

Proposition 1

$\forall s \in (0, 1)$, $\exists \tau \in (s, 1)$ and $C > 0$, such that $\forall t \in (s, \tau]$, $\int_{\mathbb{R}^{d}} |p(x_s| x_t) - \tilde{p}(x_s| x_t)| d x_s < C \sqrt{\sigma_{s|t}}$.

Figures (10)

  • Figure 1: In this paper, we explore singularities theoretically and propose a plug-and-play module SingDiffusion to address the sampling challenge at the initial singular time step. By integrating this module into existing pre-trained models, our approach effectively tackles the difficulties of generating both dark and bright images, and further enhancing overall image quality as confirmed by quantitative analysis.
  • Figure 2: Illusion of the gap between the sampling from $x_{1-\varepsilon}$ and $x_{1}$. Due to the lack of consideration of singular time step sampling in most of the existing methods, they will encounter the average brightness issue. To tackle this, we propose a plug-and-play SingDiffusion method (highlighted in red) to bridge this gap.
  • Figure 3: Comparison of stable diffusion models and SingDiffusion on average brightness issue.
  • Figure 4: Comparison of Pareto curves between SingDiffusion, SD-1.5, and SD-2.0-base on 30k COCO images, across various guidance scales in [1.5, 2, 3, 4, 5, 6, 7, 8].
  • Figure 5: Our method can be trained once and seamlessly integrated into the pre-trained models on CIVITAI in a plug-and-play fashion.
  • ...and 5 more figures

Theorems & Definitions (16)

  • Proposition 1: Error Bound Estimated by $\sigma_{s|t}$
  • Proposition 2: Error Bound Estimated by $\alpha_s$
  • Proposition 3
  • Proposition 4
  • Corollary 1
  • Corollary 2
  • Proposition 5
  • Corollary 3
  • Proposition 6
  • Lemma 1
  • ...and 6 more