Sparse Signal Reconstruction via Iterative Support Detection (0909.4359v2)

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/.

• The paper demonstrates that iterative support detection significantly reduces measurement requirements while accurately reconstructing sparse signals.
• It introduces a novel algorithm that iteratively updates all signal components using support detection and truncated l1 minimization.
• Numerical experiments show that threshold-ISD achieves superior speed and recovery performance compared to methods like IRLS and IRL1.

Sparse Signal Reconstruction via Iterative Support Detection

The paper presents a sparse signal reconstruction method known as Iterative Support Detection (ISD). This approach addresses signal reconstructions that are unsuccessful due to insufficient measurements, which the classical $\ell_1$ minimization struggles with. ISD aims for prompt signal reconstruction with fewer required measurements compared to traditional methods.

Core Concepts and Methodology

The paper establishes the foundational algorithmic structure of ISD. The core of the ISD method lies in its iterative process comprising support detection and signal reconstruction. It deviates from the traditional orthogonal matching pursuit (OMP) and its variants by not relying on a nested or expanding index set. Instead, ISD iteratively updates all components of $x$ simultaneously.

Theoretical Analysis

A key theoretical advancement within the paper is the introduction of the truncated null space property (t-NSP), a generalization of the well-known null space property (NSP). The insufficient recovery conditions of truncated $\ell_1$ minimization are investigated, which are central to explaining why ISD can often recover sparse signals when the classical methods cannot. The paper details the stability of the truncated $\ell_1$ minimization, providing bounds on the error between the true signal and the reconstructed signal under certain conditions.

Numerical Experiments and Results

Numerical experiments showcase the efficiency of ISD, specifically through its implementation variant, threshold-ISD. This version demonstrated significant improvements in performance over the classical $\ell_1$ minimization, as well as over the iterative reweighted $\ell_1$ minimization (IRL1) and the iteratively reweighted least squares (IRLS) algorithms. The results prove especially robust for signals with rapidly decaying non-zero distributions.

Threshold-ISD was successful in recovering signals with fewer measurements compared to state-of-the-art methods. The numerical validation highlights ISD’s order-of-magnitude speed retention over methods like IRLS, substantiating the practical applicability of ISD in real-world scenarios where signal dimensions are substantial.

Practical Implications and Future Directions

The practical implications of ISD and its efficient variant, threshold-ISD, suggest immediate applications in fields associated with compressive sensing and signals with fast decaying non-zero components. The findings indicate potential expansion into areas necessitating rapid, high-accuracy signal reconstruction with limited data acquisition capabilities.

The paper identifies further expansion of the ISD framework through enhanced support detection strategies accommodating diverse signal structures. Future research opportunities include developing support detection mechanisms leveraging model-based compressive sensing theories and adapting ISD to different minimization frameworks beyond $\ell_1$.

In sum, the ISD method introduces a significant advancement in sparse signal reconstruction, demonstrating improved speed and reduced measurement demands. The paper lays a formidable groundwork for future exploration into novel signal reconstruction paradigms.