  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 $\mathbf{x}$.
