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

TL;DR

This paper introduces a three-step process for sparse signal estimation that leverages a stable sparsity measure, enabling effective reconstruction without prior knowledge of the signal's support or sparsity level.

Contribution

The paper proposes a novel three-step algorithm that incorporates a stable sparsity measure into the compressed sensing framework, improving estimation without assumptions on the signal.

Findings

The method effectively estimates signals with unknown sparsity.

It requires few measurements and minimal computation.

The approach outperforms traditional methods in certain scenarios.

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 $\mathbf{x}$.