Group Symmetry Enables Faster Optimization in Inverse Problems

TL;DR

This paper proves that leveraging group symmetry in linear inverse problems allows for the design of gradient-based optimizers that achieve faster convergence, benefiting applications like medical imaging and compressed sensing.

Contribution

It introduces the first theoretical proof that symmetry structures enable the development of faster, structure-adaptive optimization algorithms for inverse problems.

Findings

Symmetry structures can be exploited for improved convergence rates.

Gradient-based optimizers can be tailored to symmetry-structured problems.

Applicable to medical imaging and compressed sensing tasks.

Abstract

We prove for the first time that, if a linear inverse problem exhibits a group symmetry structure, gradient-based optimizers can be designed to exploit this structure for faster convergence rates. This theoretical finding demonstrates the existence of a special class of structure-adaptive optimization algorithms which are tailored for symmetry-structured inverse problems such as CT/MRI/PET, compressed sensing, and image processing applications such as inpainting/deconvolution, etc.