Solve-Select-Scale: A Three Step Process For Sparse Signal Estimation
This addresses a challenge in compressed sensing for researchers and practitioners by providing a novel approach to estimate sparse signals without prior knowledge of sparsity, though it appears incremental as it builds on existing measures.
The paper tackles the problem of sparse signal estimation in compressed sensing when sparsity is unknown, by incorporating a stable measure of sparsity s(x) into the framework and proposing a three-step algorithm for solving Ax=b without additional assumptions on the signal. The result is a method that uses only a small number of linear measurements, requires little computation, and does not rely on sparsity assumptions.
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 $\mathbf{x}$.