An adaptive inverse scale space method for compressed sensing

Abstract

In this paper we introduce a novel adaptive approach for solving <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript 1"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -minimization problems as frequently arising in compressed sensing, which is based on the recently introduced inverse scale space method. The scheme allows to efficiently compute minimizers by solving a sequence of low-dimensional nonnegative least-squares problems. We provide a detailed convergence analysis in a general setup as well as refined results under special conditions. In addition, we discuss experimental observations in several numerical examples.