Regularisation, optimisation, subregularity

TL;DR

This paper advances regularisation theory in Banach spaces by employing metric subregularity to establish norm convergence for non--norm-squared regularisation, with implications for inverse problems and image reconstruction.

Contribution

It introduces a novel approach using metric subregularity to prove convergence in Banach spaces for regularisation methods, extending beyond traditional Bregman divergence techniques.

Findings

Metric subregularity ensures norm convergence in Banach spaces.

Flat regions in ground truth affect regularisation in image reconstruction.

Provides regularisation complexity bounds for optimisation algorithms.

Abstract

Regularisation theory in Banach spaces, and non--norm-squared regularisation even in finite dimensions, generally relies upon Bregman divergences to replace norm convergence. This is comparable to the extension of first-order optimisation methods to Banach spaces. Bregman divergences can, however, be somewhat suboptimal in terms of descriptiveness. Using the concept of (strong) metric subregularity, previously used to prove the fast local convergence of optimisation methods, we show norm convergence in Banach spaces and for non--norm-squared regularisation. For problems such as total variation regularised image reconstruction, the metric subregularity reduces to a geometric condition on the ground truth: flat areas in the ground truth have to compensate for the fidelity term not having second-order growth within the kernel of the forward operator. Our approach to proving such regularisation results is based on optimisation formulations of inverse problems. As a side result of the regularisation theory that we develop, we provide regularisation complexity results for optimisation methods: how many steps $N_\delta$ of the algorithm do we have to take for the approximate solutions to converge as the corruption level $\delta \searrow 0$?