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The emerging theory of compressive sensing (CS) provides a new sparse signal processing paradigm for reconstructing sparse signals from the undersampled linear measurements. Recently, numerous algorithms have been developed to solve convex optimization problems for CS sparse signal recovery. However, in some certain circumstances, greedy algorithms exhibit superior performance than convex methods. This paper is a followup to the recent paper of Wang and Yin (2010), who refine BP reconstructions via iterative support detection (ISD). The heuristic idea of ISD was applied to greedy algorithms. We developed two approaches for accelerating the ECME iteration. The described algorithms, named ECME thresholding pursuits (EMTP), introduced two greedy strategies that each iteration detects a support set

Sparsity exploiting has recently received a great amount of attention in the applied mathematics and signal processing community. Sparse signal processing algorithms have been developed for various applications. A recent Proceedings of the IEEE special issue on applications of sparse representation and compressive sensing devoted to this topic. Some of the exciting developments addressed in [

Iterative hard thresholding (IHT) [

Inspired by the theoretical and empirical evidence of favorable recovery performance of ECME [

This paper considers the iterative greedy algorithms and abstracts them into two types [

We introduce the notation used in this paper.

Qiu and Dogandzic derived an expectation conditional maximization either (ECME) iteration based on a probabilistic framework [

The algorithms described in this paper fall into the category of a general family of iterative greedy pursuit algorithms. Following [

Detect the support set

Update the signal

Many algorithms (e.g., Orthogonal Matching Pursuit (OMP) [

Considering the failed reconstructions of BP, Wang and Yin [

signal reconstruction:

solve the truncated BP with

support detection:

detect support set

The reliability of ISD relies on the support detection. Wang and Yin devised serval detection strategies for the sparse signals with components having a fast decaying distribution of nonzero components (called fast decaying signals [

In this section, we describe our proposed approaches, named ECME Thresholding Pursuits (EMTP), that combine OST and TST using the heuristic idea of ISD. The detailed description of the proposed algorithms are presented as follows

We have the following.

Initialize the reconstruction signal

initialize residual signal

initialize support set

set the iteration counter

We have the following.

Update signal approximation:

Detect the support set

strategy 1: hard thresholding

strategy 2: dynamic thresholding

We have the following.

Estimate the signal:

Update the residual:

Check the stopping condition and, if it is not yet time to stop, increment

We present two thresholding proposals and corresponding algorithm EMTP-

An EMTP-

The 1st iteration

The 2nd iteration

The 3rd iteration

The 4th iteration

The 5th iteration

The 6th iteration

Like DORE algorithm, the basic operation is matrix-vector multiplication and sorting operation. Assume that the pseudoinverse matrix

To assess the performance of the proposed approaches, we conduct numerical experiments on computer simulations. All algorithms were implemented and tested in MATLAB v7.6 running on Windows XP with 2.53 GHz Intel Celeron CPU and 2 GB of memory. We compared the proposed approaches with the accelerated ADORE/DORE approaches. The code of ADORE/DORE is available on the authors homepage (

The underlying

and the nonzeros

The sparse Gaussian signals were generated in MATLAB by

The sparse Laplacian signals were generated by

The power-law decaying signals were generated by

The variable lambda controls the rate of decay. We set

For fair comparison, we stopped iterations once the relative error fell below a certain convergence tolerance or the number of iterations is greater than 100. The convergence tolerance is given by

We evaluated the influence of thresholding parameter

Influence of thresholding parameter

Influence of thresholding parameter

Influence of thresholding parameter

Influence of thresholding parameter

We fixed the ratios of

Number of iterations as a function of problem size with fixed ratios of

Number of iterations as a function of problem size with fixed ratios of

We present comparisons in terms of SNR (dB) and number of iterations. We omit the comparisons with

Sparse Gaussian signals with

Sparse Gaussian signals with

Sparse Gaussian signals with

Sparse Gaussian signals with

As discussed in [

Sparse Laplacian signals with

Sparse Laplacian signals with

Power-law decaying signals with

Power-law decaying signals with

Probability of exact recovery for DORE, SP, and EMTP as a function of problem sparsity and four problem indeterminacies from the thickest to thinnest lines:

Following [

Comparison of phase transitions of DORE, SP, and EMTP for Gaussian-distributed sparse vectors.

To sum up, we have compared EMTP with ADORE/DORE. EMTP has significant advantages over the accelerated OST in terms of SNR and number of iterations. EMTP can significantly reduce the number of iterations required and achieves significantly higher SNR. For low indeterminacy level

In this paper, we propose ECME thresholding pursuits (EMTP) for sparse signal recovery. EMTP detects a support set

EMTP requires the precomputation and storage of the pseudoinverse matrix

In signal estimation stage, that is, update reconstructed signal by solving least-squares problem, EMTP uses the orthogonal projection, that is, by calculating

This paper devotes efforts to devise computational algorithms which are experimentally reproducible and provides empirical studies as useful guidelines for practical applications. Further, future investigations will test other OST methods as the reference and sophisticated thresholding methods for support detection.

This work was supported by the National Science Foundation of China under Grant no. 61074167, 61170126, and the Scientific Research Foundation for Advanced Talents by the Jiangsu University, no. 12JDG050.