Convexity of the ℓ1-norm based sparsity measure with respect to the missing samples as variables

Abstract

Sparse signal processing and the reconstruction of missing samples of signals exhibiting sparsity in a transform domain have been emerging research topics during the last decade. In this paper, we present the proof of the sparsity measure convexity, when considering the missing samples as minimization variables. The sparsity measure can be directly exploited in the reconstruction procedures, such as in the recently proposed gradient-based reconstruction algorithm. It makes the proof of sparsity measure convexity with respect to the missing samples as minimization variables especially interesting for signal processing. The minimal value of the sparsity measure corresponds to the set of missing sample values representing the sparsest possible solution, assuming that the reconstruction conditions are met. Convexity, along with recently presented proof of the uniqueness of the acquired solution, makes the gradient-based algorithm with missing samples as variables, a complete approach to the signal reconstruction. If the sparsity measure is convex, then we can guarantee that the solution corresponds to the global minimum of the sparsity measure, since the local minima do not exist in that case.