Sparse Regularization with $l^q$ Penalty Term

TL;DR

This paper investigates the use of Tikhonov regularization with an $l^q$ penalty to stably approximate sparse solutions of nonlinear operator equations, achieving improved convergence rates under certain conditions.

Contribution

It extends regularization theory to include $l^q$ penalties for sparse solutions, providing convergence rates and conditions for improved accuracy.

Findings

Convergence rate of $O(\sqrt{\delta})$ under standard assumptions.

Improved convergence rate of $O(\delta)$ when the solution is sparse and certain injectivity conditions hold.

Applicable to nonlinear operators with assumptions similar to the linear case.

Abstract

We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate $O(\sqrt{\delta})$ of the regularized solutions in dependence of the noise level $\delta$. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to $O(\delta)$.