Sparse Signal Reconstruction via Iterative Support Detection

TL;DR

The paper introduces ISD, a fast iterative method for sparse signal reconstruction that improves upon l_1 minimization and other algorithms by effectively estimating support sets and updating all components simultaneously.

Contribution

It proposes a novel iterative support detection method (ISD) with a generalized Null Space Property and an efficient implementation called threshold-ISD, demonstrating superior performance in experiments.

Findings

Threshold-ISD outperforms classical l_1 minimization.

Threshold-ISD surpasses IRL1 and IRLS algorithms.

ISD requires fewer measurements for accurate reconstruction.

Abstract

We present a novel sparse signal reconstruction method "ISD", aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical l_1 minimization approach. ISD addresses failed reconstructions of l_1 minimization due to insufficient measurements. It estimates a support set I from a current reconstruction and obtains a new reconstruction by solving the minimization problem \min{\sum_{i\not\in I}|x_i|:Ax=b}, and it iterates these two steps for a small number of times. ISD differs from the orthogonal matching pursuit (OMP) method, as well as its variants, because (i) the index set I in ISD is not necessarily nested or increasing and (ii) the minimization problem above updates all the components of x at the same time. We generalize the Null Space Property to Truncated Null Space Property and present our analysis of ISD based on the latter. We introduce an efficient implementation of ISD, called threshold--ISD, for recovering signals with fast decaying distributions of nonzeros from compressive sensing measurements. Numerical experiments show that threshold--ISD has significant advantages over the classical l_1 minimization approach, as well as two state--of--the--art algorithms: the iterative reweighted l_1 minimization algorithm (IRL1) and the iterative reweighted least--squares algorithm (IRLS). MATLAB code is available for download from http://www.caam.rice.edu/~optimization/L1/ISD/.