Solve-Select-Scale: A Three Step Process For Sparse Signal Estimation

Abstract: In the theory of compressed sensing (CS), the sparsity $|x|_0$ of the unknown signal $\mathbf{x} \in \mathcal{R}^n$ is of prime importance and the focus of reconstruction algorithms has mainly been either $|x|_0$ or its convex relaxation (via $|x|_1$). However, it is typically unknown in practice and has remained a challenge when nothing about the size of the support is known. As pointed recently, $|x|_0$ might not be the best metric to minimize directly, both due to its inherent complexity as well as its noise performance. Recently a novel stable measure of sparsity $s(\mathbf{x}) := |\mathbf{x}|_1^2/|\mathbf{x}|_2^2$ has been investigated by Lopes \cite{Lopes2012}, which is a sharp lower bound on $|\mathbf{x}|_0$. The estimation procedure for this measure uses only a small number of linear measurements, does not rely on any sparsity assumptions, and requires very little computation. The usage of the quantity $s(\mathbf{x})$ in sparse signal estimation problems has not received much importance yet. We develop the idea of incorporating $s(\mathbf{x})$ into the signal estimation framework. We also provide a three step algorithm to solve problems of the form $\mathbf{Ax=b}$ with no additional assumptions on the original signal $\mathbf{x}$.