Regularisation, optimisation, subregularity. (16th March 2021) - Valkonen, T
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 δ of the algorithm do we have to take for the approximate solutions to converge as the corruption level δ ↘ 0?