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

TL;DR

This paper demonstrates that score-based generative models, integrated with graduated optimisation, can effectively solve non-convex inverse problems like CT image reconstruction, converging to high-quality solutions regardless of initial conditions.

Contribution

It introduces a graduated optimisation framework using score-based generative models for inverse problems, with convergence guarantees and practical effectiveness demonstrated in CT reconstruction.

Findings

Convergence to stationary points in non-convex optimisation.

High-quality image recovery independent of initialisation.

Framework applicable to real-world inverse problems like CT.

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.