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Convergence Properties of Score-Based Models for Linear Inverse Problems Using Graduated Optimisation

Pascal Fernsel, Željko Kereta, Alexander Denker

TL;DR

The paper addresses linear inverse problems by embedding score-based generative models into a graduated optimisation framework, transforming a non-convex regularisation problem into a sequence of easier surrogates that converge to stationary points of the original objective. It introduces both a gradient-like GN flow using $F(\mathbf{x}, t)$ and an adaptive smoothing schedule, and an energy-based parametrisation to enable efficient line searches, proving convergence to stationary points under suitable conditions. Empirical results on a 2D toy problem and CT reconstruction demonstrate robust convergence and high-quality reconstructions largely independent of initialization, with adaptive step sizing yielding tangible PSNR/SSIM gains and fewer iterations. The work provides a principled blend of SGMs and graduated optimisation for practical inverse problems, with public code to foster reproducibility and further exploration.

Abstract

The incorporation of generative models as regularisers within variational formulations for inverse problems has proven effective across numerous image reconstruction tasks. However, the resulting optimisation problem is often non-convex and challenging to solve. In this work, we show that score-based generative models (SGMs) can be used in a graduated optimisation framework to solve inverse problems. We show that the resulting graduated non-convexity flow converge to stationary points of the original problem and provide a numerical convergence analysis of a 2D toy example. We further provide experiments on computed tomography image reconstruction, where we show that this framework is able to recover high-quality images, independent of the initial value. The experiments highlight the potential of using SGMs in graduated optimisation frameworks. The source code is publicly available on GitHub.

Convergence Properties of Score-Based Models for Linear Inverse Problems Using Graduated Optimisation

TL;DR

The paper addresses linear inverse problems by embedding score-based generative models into a graduated optimisation framework, transforming a non-convex regularisation problem into a sequence of easier surrogates that converge to stationary points of the original objective. It introduces both a gradient-like GN flow using and an adaptive smoothing schedule, and an energy-based parametrisation to enable efficient line searches, proving convergence to stationary points under suitable conditions. Empirical results on a 2D toy problem and CT reconstruction demonstrate robust convergence and high-quality reconstructions largely independent of initialization, with adaptive step sizing yielding tangible PSNR/SSIM gains and fewer iterations. The work provides a principled blend of SGMs and graduated optimisation for practical inverse problems, with public code to foster reproducibility and further exploration.

Abstract

The incorporation of generative models as regularisers within variational formulations for inverse problems has proven effective across numerous image reconstruction tasks. However, the resulting optimisation problem is often non-convex and challenging to solve. In this work, we show that score-based generative models (SGMs) can be used in a graduated optimisation framework to solve inverse problems. We show that the resulting graduated non-convexity flow converge to stationary points of the original problem and provide a numerical convergence analysis of a 2D toy example. We further provide experiments on computed tomography image reconstruction, where we show that this framework is able to recover high-quality images, independent of the initial value. The experiments highlight the potential of using SGMs in graduated optimisation frameworks. The source code is publicly available on GitHub.
Paper Structure (13 sections, 2 theorems, 29 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 2 theorems, 29 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Graduated Optimisation
  4. Score-based Generative Models
  5. Methods
  6. Graduated non-convexity Flow
  7. Gradient-like Methods
  8. Energy-based Parametrisation
  9. Experiments
  10. 2D Toy Example
  11. Computed Tomography
  12. Discussion
  13. Conclusion

Key Result

Theorem 3.2

Take $F$ and $f$, given in eq:f and eq:seq_minimisation_1, with $F(\cdot, t)$ continuously differentiable for $t\in [{t_\text{min}}, {t_\text{max}}]$ with ${t_\text{min}} > 0$. Assume $\{t_i\}_{i\in \mathbb{N}}$ satisfy eq:t_i. Let $\{{\mathbf{d}}_i\}_{i\in \mathbb{N}},\ \{{\mathbf{x}}_i\}_{i\in \ma

Figures (3)

  • Figure 1: Convergence properties of the graduated non-convexity flow with constant step size (Algorithm \ref{['alg:gradientLikeAlgorithm_v3']}, Figures \ref{['fig:2DIP:KoblerPock:xyplane']}-\ref{['fig:2DIP:KoblerPock:localMin']}) and the gradient-like method with the adaptive smoothing schedule (Algorithm \ref{['alg:gradientLikeAlgorithm']}, Figures \ref{['fig:2DIP:gradientLike:xyplane']}-\ref{['fig:2DIP:gradientLike:localMin']}) for the 2D toy example. We show the rate of trajectories, for different choices of ${t_\text{max}}$, converging to the global minimum depending on the initial starting point ${\mathbf{x}}_1$ (left column). Furthermore, we display the rate of trajectories which converge to the global minimum (middle column) and to stationary points (right column) depending on the initial smoothing parameter ${t_\text{max}}$ and the iteration number.
  • Figure 2: Result of the gradient like method for one example from the AAPM dataset. We show the PSNR, the objective value, the gradient-like condition and the computed step size during the iterations. The gradient-like condition is always satisfied during the iterations.
  • Figure 3: Example Reconstruction of Ellipses and AAPM. All images are shown in a color range $[0,1]$.

Theorems & Definitions (5)

  • Definition 3.1: Gradient-like Directions geiger2013numerische
  • Theorem 3.2
  • Remark 1
  • Theorem 3.2
  • proof