scholarly journals A Sharp Bound on RIC in Generalized Orthogonal Matching Pursuit

2018 ◽  
Vol 61 (1) ◽  
pp. 40-54 ◽  
Author(s):  
Wengu Chen ◽  
Huanmin Ge
Keyword(s):  
Matching Pursuit ◽  
Natural Extension ◽  
Orthogonal Matching Pursuit ◽  
Sparse Signal ◽  
Sparse Signals ◽  
Sensing Matrix ◽  
Extra Condition ◽  
Exact Reconstruction ◽  
Restricted Isometry Constant ◽  
Generalized Orthogonal

AbstractThe generalized orthogonal matching pursuit (gOMP) algorithm has received much attention in recent years as a natural extension of the orthogonal matching pursuit (OMP). It is used to recover sparse signals in compressive sensing. In this paper, a new bound is obtained for the exact reconstruction of every K-sparse signal via the gOMP algorithm in the noiseless case. That is, if the restricted isometry constant (RIC) δNK+1 of the sensing matrix A satisfiesthen the gOMP can perfectly recover every K-sparse signal x from y = Ax. Furthermore, the bound is proved to be sharp. In the noisy case, the above bound on RIC combining with an extra condition on the minimum magnitude of the nonzero components of K-sparse signals can guarantee that the gOMP selects all of the support indices of the K-sparse signals.

scholarly journals A New Generalized Orthogonal Matching Pursuit Method

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Liquan Zhao ◽  
Yulong Liu
Keyword(s):  
Matching Pursuit ◽  
High Reliability ◽  
Orthogonal Matching Pursuit ◽  
Sparse Signal ◽  
Sensing Matrix ◽  
Residual Vector ◽  
Reliability Indices ◽  
Support Set ◽  
Reconstruction Performance ◽  
Generalized Orthogonal

To improve the reconstruction performance of the generalized orthogonal matching pursuit, an improved method is proposed. Columns are selected from the sensing matrix by generalized orthogonal matching pursuit, and indices of the columns are added to the estimated support set to reconstruct a sparse signal. Those columns contain error columns that can reduce the reconstruction performance. Therefore, the proposed algorithm adds a backtracking process to remove the low-reliability columns from the selected column set. For any k-sparse signal, the proposed method firstly computes the correlation between the columns of the sensing matrix and the residual vector and then selects s columns that correspond to the s largest correlation in magnitude and adds their indices to the estimated support set in each iteration. Secondly, the proposed algorithm projects the measurements onto the space that consists of those selected columns and calculates the projection coefficient vector. When the size of the support set is larger than k, the proposed method will select k high-reliability indices using a search strategy from the support set. Finally, the proposed method updates the estimated support set using the selected k high-reliability indices. The simulation results demonstrate that the proposed algorithm has a better recovery performance.

scholarly journals A Sharp RIP Condition for Orthogonal Matching Pursuit

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Wei Dan
Keyword(s):  
Matching Pursuit ◽  
Restricted Isometry Property ◽  
Orthogonal Matching Pursuit ◽  
Sparse Signal ◽  
Sparse Signals

A restricted isometry property (RIP) conditionδK+KθK,1<1is known to be sufficient for orthogonal matching pursuit (OMP) to exactly recover everyK-sparse signalxfrom measurementsy=Φx. This paper is devoted to demonstrate that this condition is sharp. We construct a specific matrix withδK+KθK,1=1such that OMP cannot exactly recover someK-sparse signals.

scholarly journals A New Analysis for Support Performance with Block Generalized Orthogonal Matching Pursuit

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
C. Y. Xia ◽  
Z. L. Zhou ◽  
Chun-Bo Guo ◽  
Y. S. Hao ◽  
C. B. Hou
Keyword(s):  
Compressive Sensing ◽  
Matching Pursuit ◽  
Restricted Isometry Property ◽  
Orthogonal Matching Pursuit ◽  
Sparse Signals ◽  
Support Conditions ◽  
Generalized Orthogonal ◽  
Recovery Block ◽  
Block Structures

For recovering block-sparse signals with unknown block structures using compressive sensing, a block orthogonal matching pursuit- (BOMP-) like block generalized orthogonal matching pursuit (BgOMP) algorithm has been proposed recently. This paper focuses on support conditions of recovery of any K -sparse block signals incorporating BgOMP under the framework of restricted isometry property (RIP). The proposed support conditions guarantee that BgOMP can achieve accurate recovery block-sparse signals within k iterations.

Perturbation Analysis of Orthogonal Least Squares

2019 ◽  
Vol 62 (4) ◽  
pp. 780-797 ◽  
Author(s):  
Pengbo Geng ◽  
Wengu Chen ◽  
Huanmin Ge
Keyword(s):  
Least Squares ◽  
Restricted Isometry Property ◽  
The Other ◽  
Sparse Signal ◽  
Sparse Signals ◽  
Sensing Matrix ◽  
Recovery Algorithm ◽  
Orthogonal Least Squares ◽  
Restricted Isometry Constant ◽  
Term Approximation

AbstractThe Orthogonal Least Squares (OLS) algorithm is an efficient sparse recovery algorithm that has received much attention in recent years. On one hand, this paper considers that the OLS algorithm recovers the supports of sparse signals in the noisy case. We show that the OLS algorithm exactly recovers the support of $K$-sparse signal $\boldsymbol{x}$ from $\boldsymbol{y}=\boldsymbol{\unicode[STIX]{x1D6F7}}\boldsymbol{x}+\boldsymbol{e}$ in $K$ iterations, provided that the sensing matrix $\boldsymbol{\unicode[STIX]{x1D6F7}}$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\unicode[STIX]{x1D6FF}_{K+1}<1/\sqrt{K+1}$, and the minimum magnitude of the nonzero elements of $\boldsymbol{x}$ satisfies some constraint. On the other hand, this paper demonstrates that the OLS algorithm exactly recovers the support of the best $K$-term approximation of an almost sparse signal $\boldsymbol{x}$ in the general perturbations case, which means both $\boldsymbol{y}$ and $\boldsymbol{\unicode[STIX]{x1D6F7}}$ are perturbed. We show that the support of the best $K$-term approximation of $\boldsymbol{x}$ can be recovered under reasonable conditions based on the restricted isometry property (RIP).

A perturbation analysis based on group sparse representation with orthogonal matching pursuit

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Chunyan Liu ◽  
Feng Zhang ◽  
Wei Qiu ◽  
Chuan Li ◽  
Zhenbei Leng
Keyword(s):  
Numerical Study ◽  
Matching Pursuit ◽  
Sufficient Conditions ◽  
Schur Complement ◽  
Orthogonal Matching Pursuit ◽  
Sparse Signal ◽  
Sparse Signals ◽  
Actual Performance ◽  
Sharp Condition ◽  
Total Noise

AbstractIn this paper, by exploiting orthogonal projection matrix and block Schur complement, we extend the study to a complete perturbation model. Based on the block-restricted isometry property (BRIP), we establish some sufficient conditions for recovering the support of the block 𝐾-sparse signals via block orthogonal matching pursuit (BOMP) algorithm. Under some constraints on the minimum magnitude of the nonzero elements of the block 𝐾-sparse signals, we prove that the support of the block 𝐾-sparse signals can be exactly recovered by the BOMP algorithm in the case of \ell_{2} and \ell_{2}/\ell_{\infty} bounded total noise if 𝑨 satisfies the BRIP of order K+1 with\delta_{K+1}<\frac{1}{\sqrt{K+1}(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}+\frac{1}{(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}-1.In addition, we also show that this is a sharp condition for exactly recovering any block 𝐾-sparse signal with the BOMP algorithm. Moreover, we also give the reconstruction upper bound of the error between the recovered block-sparse signal and the original block-sparse signal. In the noiseless and perturbed case, we also prove that the BOMP algorithm can exactly recover the block 𝐾-sparse signal under some constraints on the block 𝐾-sparse signal and\delta_{K+1}<\frac{2+\sqrt{2}}{2(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}-1.Finally, we compare the actual performance of perturbed OMP and perturbed BOMP algorithm in the numerical study. We also present some numerical experiments to verify the main theorem by using the completely perturbed BOMP algorithm.

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