Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

TL;DR

This paper investigates convergence rates for Tikhonov regularization with sparsity constraints using weighted ll^p penalties, establishing new results for different p values and highlighting the importance of sparsity and operator interplay.

Contribution

It provides new convergence rate results for ll^p regularization with sparsity constraints, especially for p=1, and discusses the limitations for p<1.

Findings

Convergence rate of ll^p regularization is  for sparse solutions when 1.

Special techniques are needed for p=1, involving Bregman and Bregman-Taylor distances.

Regularization may fail for p=0, depending on the operator and basis.

Abstract

This paper addresses the regularization by sparsity constraints by means of weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main result it is proven that one gets a convergence rate in norm of $\sqrt{\delta}$ for $1\leq p\leq 2$ as soon as the unknown solution is sparse. The case $p=1$ needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed. For $p<1$ only preliminary results are shown. These results indicate that, different from $p\geq 1$, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for $p=0$ shows that regularization need not to happen.