Convergence Rates for Oversmoothing Banach Space Regularization

TL;DR

This paper develops a new approach using $K$-interpolation theory to analyze convergence rates in Banach space regularization, achieving optimal rates for various penalties and noise models with simplified proofs.

Contribution

It introduces a novel $K$-interpolation based method for establishing convergence rates in oversmoothing Banach space regularization, extending results to new penalty types.

Findings

Achieves optimal convergence rates for total variation and Besov penalties.

Demonstrates minimax optimal rates in white noise models.

Provides numerical validation of theoretical results.

Abstract

This paper studies Tikhonov regularization for finitely smoothing operators in Banach spaces when the penalization enforces too much smoothness in the sense that the penalty term is not finite at the true solution. In a Hilbert space setting, Natterer (1984) showed with the help of spectral theory that optimal rates can be achieved in this situation. ('Oversmoothing does not harm.') For oversmoothing variational regularization in Banach spaces only very recently progress has been achieved in several papers on different settings, all of which construct families of smooth approximations to the true solution. In this paper we propose to construct such a family of smooth approximations based on $K$-interpolation theory. We demonstrate that this leads to simple, self-contained proofs and to rather general results. In particular, we obtain optimal convergence rates for bounded variation regularization, general Besov penalty terms and $\ell^p$ wavelet penalization with $p<1$ which cannot be treated by previous approaches. We also derive minimax optimal rates for white noise models. Our theoretical results are confirmed in numerical experiments.