Stable reconstructions for the analysis formulation of $\ell^p$-minimization using redundant systems

TL;DR

This paper investigates the stability of $ ext{l}^p$-minimization solutions in compressed sensing using redundant, possibly non-tight frames, introducing a new concept of frames with identifiable duals and highlighting gaps between theory and practice.

Contribution

It introduces the concept of frames with identifiable duals for stability analysis and extends $ ext{l}^p$-minimization stability results to general redundant transforms beyond tight frames.

Findings

Stability results for $ ext{l}^p$-minimization with redundant frames.

Identification of a gap between theoretical guarantees and practical applications.

Numerical experiments illustrating the theoretical insights.

Abstract

In compressed sensing sparse solutions are usually obtained by solving an $\ell^1$-minimization problem. Furthermore, the sparsity of the signal does need not be directly given. In fact, it is sufficient to have a signal that is sparse after an application of a suitable transform. In this paper we consider the stability of solutions obtained from $\ell^p$-minimization for arbitrary $0<p \leq1$. Further we suppose that the signals are sparse with respect to general redundant transforms associated to not necessarily tight frames. Since we are considering general frames the role of the dual frame has to be additionally discussed. For our stability analysis we will introduce a new concept of so-called \emph{frames with identifiable duals}. Further, we numerically highlight a gap between the theory and the applications of compressed sensing for some specific redundant transforms.