On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization

TL;DR

This paper investigates conditions under which sparsity regularization methods in ill-posed problems achieve convergence, showing that certain restrictive range conditions are always satisfied under weaker topological assumptions, with applications to classical integral operators.

Contribution

It demonstrates that the range condition for basis elements in $ ext{ell}^1$-regularization is always met under weaker topology assumptions, broadening applicability.

Findings

Range condition always satisfied under weaker topology

Applicable to Radon transform and Volterra operators

Extends to non-convex $ ext{ell}^q$-regularization with 0<q<1

Abstract

The convergence rates results in $\ell^1$-regularization when the sparsity assumption is narrowly missed, presented by Burger et al. (2013 Inverse Problems 29 025013), are based on a crucial condition which requires that all basis elements belong to the range of the adjoint of the forward operator. Partly it was conjectured that such a condition is very restrictive. In this context, we study sparsity-promoting varieties of Tikhonov regularization for linear ill-posed problems with respect to an orthonormal basis in a separable Hilbert space using $\ell^1$ and sublinear penalty terms. In particular, we show that the corresponding range condition is always satisfied for all basis elements if the problems are well-posed in a certain weaker topology and the basis elements are chosen appropriately related to an associated Gelfand triple. The Radon transform, Symm's integral equation and linear integral operators of Volterra type are examples for such behaviour, which allows us to apply convergence rates results for non-sparse solutions, and we further extend these results also to the case of non-convex $\ell^q$-regularization with 0<q<1.