Convergence rates in $\ell^1$-regularization when the basis is not smooth enough

TL;DR

This paper extends convergence rate analysis in -regularization to non-sparse solutions, providing error estimates under weaker assumptions and demonstrating broader applicability in practical scenarios.

Contribution

It introduces error estimates for -regularization without requiring the solution to be sparse, under weaker assumptions than previous work.

Findings

Error estimates are valid for non-sparse solutions.

Weaker assumptions expand applicability to real-world problems.

Examples demonstrate the relevance of the new assumptions.

Abstract

Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation with respect to a fixed basis. We drop this sparsity assumption and provide error estimates for non-sparse solutions. After discussing a result in this direction published earlier by one of the authors and coauthors we prove a similar error estimate under weaker assumptions. Two examples illustrate that this set of weaker assumptions indeed covers additional situations which appear in applications.