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Stability and Generalizability in SDE Diffusion Models with Measure-Preserving Dynamics

Weitong Zhang, Chengqi Zang, Liu Li, Sarah Cechnicka, Cheng Ouyang, Bernhard Kainz

TL;DR

This work uncovers several strategies that inherently enhance the stability and generalizability of diffusion models for inverse problems and introduces a novel score-based diffusion framework, the D$3$GM, which can return the degraded measurement to the original state despite complex degradation with the RDS concept of stability.

Abstract

Inverse problems describe the process of estimating the causal factors from a set of measurements or data. Mapping of often incomplete or degraded data to parameters is ill-posed, thus data-driven iterative solutions are required, for example when reconstructing clean images from poor signals. Diffusion models have shown promise as potent generative tools for solving inverse problems due to their superior reconstruction quality and their compatibility with iterative solvers. However, most existing approaches are limited to linear inverse problems represented as Stochastic Differential Equations (SDEs). This simplification falls short of addressing the challenging nature of real-world problems, leading to amplified cumulative errors and biases. We provide an explanation for this gap through the lens of measure-preserving dynamics of Random Dynamical Systems (RDS) with which we analyse Temporal Distribution Discrepancy and thus introduce a theoretical framework based on RDS for SDE diffusion models. We uncover several strategies that inherently enhance the stability and generalizability of diffusion models for inverse problems and introduce a novel score-based diffusion framework, the \textbf{D}ynamics-aware S\textbf{D}E \textbf{D}iffusion \textbf{G}enerative \textbf{M}odel (D$^3$GM). The \textit{Measure-preserving property} can return the degraded measurement to the original state despite complex degradation with the RDS concept of \textit{stability}. Our extensive experimental results corroborate the effectiveness of D$^3$GM across multiple benchmarks including a prominent application for inverse problems, magnetic resonance imaging. Code and data will be publicly available.

Stability and Generalizability in SDE Diffusion Models with Measure-Preserving Dynamics

TL;DR

This work uncovers several strategies that inherently enhance the stability and generalizability of diffusion models for inverse problems and introduces a novel score-based diffusion framework, the DGM, which can return the degraded measurement to the original state despite complex degradation with the RDS concept of stability.

Abstract

Inverse problems describe the process of estimating the causal factors from a set of measurements or data. Mapping of often incomplete or degraded data to parameters is ill-posed, thus data-driven iterative solutions are required, for example when reconstructing clean images from poor signals. Diffusion models have shown promise as potent generative tools for solving inverse problems due to their superior reconstruction quality and their compatibility with iterative solvers. However, most existing approaches are limited to linear inverse problems represented as Stochastic Differential Equations (SDEs). This simplification falls short of addressing the challenging nature of real-world problems, leading to amplified cumulative errors and biases. We provide an explanation for this gap through the lens of measure-preserving dynamics of Random Dynamical Systems (RDS) with which we analyse Temporal Distribution Discrepancy and thus introduce a theoretical framework based on RDS for SDE diffusion models. We uncover several strategies that inherently enhance the stability and generalizability of diffusion models for inverse problems and introduce a novel score-based diffusion framework, the \textbf{D}ynamics-aware S\textbf{D}E \textbf{D}iffusion \textbf{G}enerative \textbf{M}odel (DGM). The \textit{Measure-preserving property} can return the degraded measurement to the original state despite complex degradation with the RDS concept of \textit{stability}. Our extensive experimental results corroborate the effectiveness of DGM across multiple benchmarks including a prominent application for inverse problems, magnetic resonance imaging. Code and data will be publicly available.
Paper Structure (21 sections, 5 theorems, 61 equations, 10 figures, 11 tables)

This paper contains 21 sections, 5 theorems, 61 equations, 10 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Instability Analysis: Transitionary SGMs with Corrupted SDE Diffusion
  4. Towards Stability: Measure-preserving Dynamics in SDE Diffusion
  5. Experiments
  6. Stability: Illustrations of the Measure-preserving Dynamics within Diffusion Models
  7. Generalizability: D$^3$GM for real-world data
  8. Discussion and limitations
  9. Conclusion
  10. Broader Impacts
  11. Intuition of Measure-Preserving Dynamics in SDE:
  12. Mathematical Preliminaries
  13. Preliminaries and Proof for Proposition 1
  14. Proof for reverse-time D$^3$GM process:
  15. Analysis of Proposition 1
  16. ...and 6 more sections

Key Result

Proposition 1

After extending the solution of the OU process to RDS, the measure-preserving RDS $\varphi$ should meet the property $\varphi(t, s ; \omega) x=\varphi\left(t-s, 0 ; \vartheta_s \omega\right) x$. However, OU processes with time-varying coefficients usually do not satisfy this property. In this situat

Figures (10)

  • Figure 1: Dynamics-aware SDE Diffusion Generative Model (D$^3$GM). When extending transitionary SDEs to random dynamical systems (RDS), their measure-preserving property should be kept to maintain stability. This corresponds to driving the SDE towards the drift term $\mu$ (LQ). There is a Temporal Distribution Discrepancy which results from the gap between the forward estimation $x_T$ and the low quality image in the SDE. With the distribution aligned between $x_T$ and $\mu$, the SDE can be made more robust to inverse problems. Reconstruction results for low quality (LQ) images after application of our D$^3$GM method, on different tasks, compared to the ground truth (GT) on two domains - The frequency domain: MRI Reconstruction (undersampling factor 8x, 16x, frequency masks are colored red); MRI Super-resolution (up-scaling factor of X4, cross-domain evaluation). The image domain: Real Dense Haze Removal; Rain Removal (light, heavy).
  • Figure 2: Qualitative results for (a) deraining and (b) dehazing.
  • Figure 3: Sampling trajectories of SGM, transitionary SGMs: Coef. Dec., OU SDE, and D$^3$GM.
  • Figure 4: Reverse Initialization with Basin of attraction.
  • Figure 5: Deraining results with light rain images of our method.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Proposition 1
  • Proposition 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Definition 5
  • Definition 6
  • Theorem 2
  • ...and 4 more