Tikhonov regularization with l^0-term complementing a convex penalty: l^1 convergence under sparsity constraints

TL;DR

This paper investigates an l^0-regularization method combined with convex penalties for solving ill-posed operator equations, providing error estimates, convergence rates, and numerical validation for sparse solutions.

Contribution

It introduces an analysis of l^0-regularization with convex penalties, deriving convergence rates and error estimates under sparsity assumptions, and explores the balance between l^0 and convex terms.

Findings

Error estimates and convergence rates are established for sparse solutions.

Explicit dependence between l^0-term and convex penalty is derived.

Numerical experiments confirm the sparsity-promoting properties of the method.

Abstract

Measuring the error by an l^1-norm, we analyze under sparsity assumptions an l^0-regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations Ax = y with an injective and bounded linear operator A mapping between l^2 and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the l^0-term and the complementing convex penalty, the important special case of the l^2-norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.