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2018, 12(1): 153-174. doi: 10.3934/ipi.2018006

Recovery of block sparse signals under the conditions on block RIC and ROC by BOMP and BOMMP

1. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

2. 

Graduate School, China Academy of Engineering Physics, Beijing 100088, China

* Corresponding author: Wengu Chen

Received  October 2016 Revised  September 2017 Published  December 2017

Fund Project: The first author is supported by NSF of China grant 11271050,11371183

In this paper, we consider the block orthogonal matching pursuit (BOMP) algorithm and the block orthogonal multi-matching pursuit (BOMMP) algorithm respectively to recover block sparse signals from an underdetermined system of linear equations. We first introduce the notion of block restricted orthogonality constant (ROC), which is a generalization of the standard restricted orthogonality constant, and establish respectively the sufficient conditions in terms of the block RIC and ROC to ensure the exact and stable recovery of any block sparse signals in both noiseless and noisy cases through the BOMP and BOMMP algorithm. We finally show that the sufficient condition on the block RIC and ROC is sharp for the BOMP algorithm.

Citation: Wengu Chen, Huanmin Ge. Recovery of block sparse signals under the conditions on block RIC and ROC by BOMP and BOMMP. Inverse Problems & Imaging, 2018, 12 (1) : 153-174. doi: 10.3934/ipi.2018006
References:
[1]

R. Baraniuk, P. Steeghs, Compressive radar imaging, in Proc. IEEE Radar Conf., 303 (2007), 128-133. doi: 10.1109/RADAR.2007.374203.

[2]

T. Blumensath, M. E. Davies, Iterative thresholding for sparse approximations, J. Fourier Anal. Appl., 14 (2008), 629-654. doi: 10.1007/s00041-008-9035-z.

[3]

T. Blumensath, M. E. Davies, Iterative hard thresholding for compressed sensing, Appl. Comput. Harmon. Anal., 27 (2008), 265-274. doi: 10.1016/j.acha.2009.04.002.

[4]

T. T. Cai, A. Zhang, Compressed sensing and affine rank minimization under restricted isometry, IEEE Trans. Signal Process., 61 (2013), 3279-3290. doi: 10.1109/TSP.2013.2259164.

[5]

T. T. Cai, A. Zhang, Sparse representation of a polytope and recovery of sparse signals and low-rank matrices, IEEE Trans. Inf. Theory, 60 (2014), 122-132. doi: 10.1109/TIT.2013.2288639.

[6]

E. J. Candés, M. B. Wakin, S. P. Boyd, Enhancing sparsity by reweighted $\ell_1 $ minimization, J. Fourier Anal. Appl., 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x.

[7]

E. J. Candés, J. Romberg, T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.

[8]

E. J. Candés, T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979.

[9]

J. Chen, X. Huo, Theoretical results on sparse representations of multiple-measurement vectors, IEEE Trans. Signal Process., 54 (2006), 4634-4643. doi: 10.1109/TSP.2006.881263.

[10]

S. Chen, D. L. Donoho, M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), 33-61. doi: 10.1137/S1064827596304010.

[11]

W. G. Chen and H. M. Ge, A sharp bound on RIC in generalize orthogonal matching pursuit, Canadian Mathematical Bulletin. doi: 10.4153/CMB-2017-009-6.

[12]

W. G. Chen, H. M. Ge, A sharp recovery condition for block sparse signals by block orthogonal multi-matching pursuit, Sci. China Math., 60 (2017), 1325-1340. doi: 10.1007/s11425-016-0448-7.

[13]

S. F. Cotter, B. D. Rao, K. Engan, K. Kreutz-Delgado, Sparse solutions to linear inverse problems with multiple measurement vectors, IEEE Trans. Signal Process., 53 (2005), 2477-2488. doi: 10.1109/TSP.2005.849172.

[14]

W. Dai, O. Milenkovic, Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory, 55 (2009), 2230-2249. doi: 10.1109/TIT.2009.2016006.

[15]

W. Dan, R. H. Wang, Robustness of orthogonal matching pursuit under restricted isometry property, Sci. China, Math., 57 (2014), 627-634. doi: 10.1007/s11425-013-4655-4.

[16]

W. Dan, Analysis of orthogonal multi-matching pursuit under restricted isometry property, Sci. China, Math., 57 (2014), 2179-2188. doi: 10.1007/s11425-014-4843-x.

[17]

W. Dan, A sharp RIP condition for orthogonal matching pursuit Abstr. Appl. Anal. 2013 (2013), Article ID 482357, 3 pages. doi: 10.1155/2013/482357.

[18]

I. Daubechies, M. Defrise, C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.

[19]

M. A. Davenport, M. B. Wakin, Analysis of orthogonal matching pursuit using the restricted isometry property, IEEE Trans. Inf. Theory, 56 (2010), 4395-4401. doi: 10.1109/TIT.2010.2054653.

[20]

D. L. Donoho, I. Drori, Y. Tsaig, J. L. Starck, Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit, IEEE Trans. Inf. Theory, 58 (2012), 1094-1121. doi: 10.1109/TIT.2011.2173241.

[21]

D. L. Donoho, Denoising by soft-threshold, IEEE Trans. Inf. Theory, 41 (1995), 613-627. doi: 10.1109/18.382009.

[22]

D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[23]

D. L. Donoho, X. Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theory, 47 (2001), 2845-2862. doi: 10.1109/18.959265.

[24]

Y. C. Eldar, P. Kuppinger, H. Bölcskei, Block-sparse signals: uncertainty relations and efficient recovery, IEEE Trans. Signal Process., 58 (2010), 3042-3054. doi: 10.1109/TSP.2010.2044837.

[25]

Y. C. Eldar, M. Mishali, Robust recovery of signals from a structured union of subspaces, IEEE Trans. Inf. Theory, 55 (2009), 5302-5316. doi: 10.1109/TIT.2009.2030471.

[26]

Y. C. Eldar, M. Mishali, Block-sparsity and sampling over a union of subspaces, In Pro. 16th Int. Conf. Digital Signal Processing, (2009), 1-8. doi: 10.1109/ICDSP.2009.5201211.

[27]

Y. L. Fu, H. F. Li, Q. H. Zhang, H. Zou, Block-sparse recovery via redundant block OMP, Signal Process., 97 (2014), 162-171. doi: 10.1016/j.sigpro.2013.10.030.

[28]

B. X. Huang, T. Zhou, Recovery of block sparse signals by a block version of StOMP, Signal Process., 106 (2015), 231-244. doi: 10.1016/j.sigpro.2014.07.023.

[29]

J. Huang, T. Zhang, The benefit of group sparsity, Ann. Stat., 38 (2010), 1978-2004. doi: 10.1214/09-AOS778.

[30]

J. H. Lin, S. Li, Block sparse recovery via mixed $\ell_2/\ell_1 $ minimization, Acta Math. Sin., 29 (2013), 1401-1412. doi: 10.1007/s10114-013-1564-y.

[31]

E. Liu, V. N. Temlyakov, The orthogonal super greedy algorithm and applications in compressed sensing, IEEE Trans. Inf. Theory, 58 (2012), 2040-2047. doi: 10.1109/TIT.2011.2177632.

[32]

M. Lustig, D. L. Donoho, J. M. Santos, J. M. Pauly, Compressed sensing MRI, IEEE Signal Process. Mag., 27 (2008), 72-82.

[33]

A. Majumdar, R. K. Ward, Compressed sensing of color images, Signal Process., 90 (2010), 3122-3127. doi: 10.1016/j.sigpro.2010.05.016.

[34]

M. Mishali, Y. C. Eldar, Blind multi-band signal reconstruction: Compressed sensing for analog signals, IEEE Trans. Signal Process., 57 (2009), 993-1009. doi: 10.1109/TSP.2009.2012791.

[35]

M. Mishali, Y. C. Eldar, Reduce and boost: Recovering arbitrary sets of jointly sparse vectors, IEEE Trans. Signal Process., 56 (2008), 4692-4702. doi: 10.1109/TSP.2008.927802.

[36]

Q. Mo, Y. Shen, A remark on the restricted isometry property in orthogonal matching pursuit, IEEE Trans. Inf. Theory, 58 (2012), 3654-3656. doi: 10.1109/TIT.2012.2185923.

[37]

Q. Mo, A sharp restricted isometry constant bound of orthogonal matching pursuit, arXiv: 1501.01708.

[38]

D. Needell, J. A. Troop, CoSaMP: Itertive signal recovery from incomplete and inaccurate samples, Appl. Comput. Harmon. Anal., 26 (2009), 301-321. doi: 10.1016/j.acha.2008.07.002.

[39]

D. Needell, R. Vershynin, Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit, IEEE J. Sel. Top. Signal Process., 4 (2010), 310-316. doi: 10.1109/JSTSP.2010.2042412.

[40]

F. Parvaresh, H. Vikalo, S. Misra, B. Hassibi, Recovering sparse signals using sparse measurement matrices incompressed DNA microarrays, IEEE J. Sel. Top. Signal Process., 2 (2008), 275-285.

[41]

B. D. Rao, K. Kreutz-Delgado, An affine scaling methodology for best basis selection, IEEE Trans. Signal Process., 47 (1999), 187-200. doi: 10.1109/78.738251.

[42]

S. Satpathi, R. L. Das, M. Chakraborty, Improving the bound on the RIP constant in generalized orthogonal matching pursuit, IEEE Signal Proc. Lett., 20 (2013), 1074-1077. doi: 10.1109/LSP.2013.2279977.

[43]

Y. Shen, S. Li, Sparse signals recovery from noisy measurements by orthogonal matching pursuit, Inverse Problems and Imaging, 9 (2015), 231-238. doi: 10.3934/ipi.2015.9.231.

[44]

G. Swirszcz, N. Abe, A. C. Lozano, Grouped orthogonal matching pursuit for variable selection and prediction, Advances in Neural Information Processing Systems, (2009), 1150-1158.

[45]

J. A. Tropp, Greed is Good: Algorithmic results for sparse approximation, IEEE Trans. Inf. Theory, 50 (2004), 2231-2242. doi: 10.1109/TIT.2004.834793.

[46]

J. A. Tropp, Algorithms for simultaneous sparse approximation. Part Ⅰ: Greedy pursuit, Signal Process., 86 (2006), 572-588. doi: 10.1016/j.sigpro.2005.05.030.

[47]

J. A. Tropp, A. C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inf. Theory, 53 (2007), 4655-4666. doi: 10.1109/TIT.2007.909108.

[48]

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, R. G. Baraniuk, Beyond Nyquist: Efficient sampling of sparse bandlimited signals, IEEE Trans. Inf. Theory, 56 (2010), 520-544. doi: 10.1109/TIT.2009.2034811.

[49]

J. Wang, B. Shim, On the recovery limit of sparse signals using orthogonal matching pursuit, IEEE Trans. Signal Process., 60 (2012), 4973-4976. doi: 10.1109/TSP.2012.2203124.

[50]

J. Wang, S. Kwon, B. Shim, Generalized orthogonal matching pursuit, IEEE Trans. Signal Process., 60 (2012), 6202-6216. doi: 10.1109/TSP.2012.2218810.

[51]

Y. Wang, J. J. Wang, Z. B. Xu, Restricted $p$-isometry properties of nonconvex block-sparse compressed sensing, Signal Process., 104 (2014), 188-196. doi: 10.1016/j.sigpro.2014.03.040.

[52]

Y. Wang, J. J. Wang, Z. B. Xu, On recovery of block-sparse signals via mixed $\ell_2/\ell_p(0 < p≤q1) $ norm minimization, EURASIP J. Adv. Signal Process., 76 (2013), 1-17.

[53]

J. M. Wen, Z. C. Zhou, Z. L. Liu, M. J. Lai and X. H. Tang, Sharp sufficient conditions for stable recovery of block sparse signals by block orthogonal matching pursuit, arXiv: 1605.02894.

[54]

J. M. Wen, Z. C. Zhou, D. F. Li, X. H. Tang, A novel sufficient condition for generalized orthogonal matching pursuit, IEEE Communications Letters, 21 (2017), 805-808. doi: 10.1109/LCOMM.2016.2642922.

[55]

J. M. Wen, Z. C. Zhou, J. Wang, X. H. Tang, Q. Mo, A sharp condition for exact support recovery with orthogonal matching pursuit, IEEE Trans. Signal Process., 65 (2017), 1370-1382. doi: 10.1109/TSP.2016.2634550.

[56]

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Anal. Mach. Intell., 31 (2009), 210-227. doi: 10.1109/AFGR.2008.4813404.

[57]

R. Wu, W. Huang, D. R. Chen, The exact support recovery of sparse signals with noise via orthogonal matching pursuit, IEEE Signal Proc. Let., 20 (2013), 403-406.

[58]

Y. Xu, X. H. Qiu, Block-sparse signals recovery using orthogonal multimatching (Chinese), Journal of Signal Processing, 30 (2014), 706-711.

[59]

Z. Q. Xu, The performance of orthogonal multi-matching pursuit under RIP, J. Comp. Math. Sci., 33 (2015), 495-516. doi: 10.4208/jcm.1505-m4529.

[60]

Q. Zhao, J. K. Wang, Y. H. Han, P. Han, Compressive sensing of block-sparse signals recovery based on sparsity adaptive regularized orthogonal matching pursuit algorithm, IEEE Fifth International Conference on Advanced Computational Intelligence, (2012), 1141-1144.

show all references

References:
[1]

R. Baraniuk, P. Steeghs, Compressive radar imaging, in Proc. IEEE Radar Conf., 303 (2007), 128-133. doi: 10.1109/RADAR.2007.374203.

[2]

T. Blumensath, M. E. Davies, Iterative thresholding for sparse approximations, J. Fourier Anal. Appl., 14 (2008), 629-654. doi: 10.1007/s00041-008-9035-z.

[3]

T. Blumensath, M. E. Davies, Iterative hard thresholding for compressed sensing, Appl. Comput. Harmon. Anal., 27 (2008), 265-274. doi: 10.1016/j.acha.2009.04.002.

[4]

T. T. Cai, A. Zhang, Compressed sensing and affine rank minimization under restricted isometry, IEEE Trans. Signal Process., 61 (2013), 3279-3290. doi: 10.1109/TSP.2013.2259164.

[5]

T. T. Cai, A. Zhang, Sparse representation of a polytope and recovery of sparse signals and low-rank matrices, IEEE Trans. Inf. Theory, 60 (2014), 122-132. doi: 10.1109/TIT.2013.2288639.

[6]

E. J. Candés, M. B. Wakin, S. P. Boyd, Enhancing sparsity by reweighted $\ell_1 $ minimization, J. Fourier Anal. Appl., 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x.

[7]

E. J. Candés, J. Romberg, T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.

[8]

E. J. Candés, T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979.

[9]

J. Chen, X. Huo, Theoretical results on sparse representations of multiple-measurement vectors, IEEE Trans. Signal Process., 54 (2006), 4634-4643. doi: 10.1109/TSP.2006.881263.

[10]

S. Chen, D. L. Donoho, M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), 33-61. doi: 10.1137/S1064827596304010.

[11]

W. G. Chen and H. M. Ge, A sharp bound on RIC in generalize orthogonal matching pursuit, Canadian Mathematical Bulletin. doi: 10.4153/CMB-2017-009-6.

[12]

W. G. Chen, H. M. Ge, A sharp recovery condition for block sparse signals by block orthogonal multi-matching pursuit, Sci. China Math., 60 (2017), 1325-1340. doi: 10.1007/s11425-016-0448-7.

[13]

S. F. Cotter, B. D. Rao, K. Engan, K. Kreutz-Delgado, Sparse solutions to linear inverse problems with multiple measurement vectors, IEEE Trans. Signal Process., 53 (2005), 2477-2488. doi: 10.1109/TSP.2005.849172.

[14]

W. Dai, O. Milenkovic, Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory, 55 (2009), 2230-2249. doi: 10.1109/TIT.2009.2016006.

[15]

W. Dan, R. H. Wang, Robustness of orthogonal matching pursuit under restricted isometry property, Sci. China, Math., 57 (2014), 627-634. doi: 10.1007/s11425-013-4655-4.

[16]

W. Dan, Analysis of orthogonal multi-matching pursuit under restricted isometry property, Sci. China, Math., 57 (2014), 2179-2188. doi: 10.1007/s11425-014-4843-x.

[17]

W. Dan, A sharp RIP condition for orthogonal matching pursuit Abstr. Appl. Anal. 2013 (2013), Article ID 482357, 3 pages. doi: 10.1155/2013/482357.

[18]

I. Daubechies, M. Defrise, C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.

[19]

M. A. Davenport, M. B. Wakin, Analysis of orthogonal matching pursuit using the restricted isometry property, IEEE Trans. Inf. Theory, 56 (2010), 4395-4401. doi: 10.1109/TIT.2010.2054653.

[20]

D. L. Donoho, I. Drori, Y. Tsaig, J. L. Starck, Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit, IEEE Trans. Inf. Theory, 58 (2012), 1094-1121. doi: 10.1109/TIT.2011.2173241.

[21]

D. L. Donoho, Denoising by soft-threshold, IEEE Trans. Inf. Theory, 41 (1995), 613-627. doi: 10.1109/18.382009.

[22]

D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[23]

D. L. Donoho, X. Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theory, 47 (2001), 2845-2862. doi: 10.1109/18.959265.

[24]

Y. C. Eldar, P. Kuppinger, H. Bölcskei, Block-sparse signals: uncertainty relations and efficient recovery, IEEE Trans. Signal Process., 58 (2010), 3042-3054. doi: 10.1109/TSP.2010.2044837.

[25]

Y. C. Eldar, M. Mishali, Robust recovery of signals from a structured union of subspaces, IEEE Trans. Inf. Theory, 55 (2009), 5302-5316. doi: 10.1109/TIT.2009.2030471.

[26]

Y. C. Eldar, M. Mishali, Block-sparsity and sampling over a union of subspaces, In Pro. 16th Int. Conf. Digital Signal Processing, (2009), 1-8. doi: 10.1109/ICDSP.2009.5201211.

[27]

Y. L. Fu, H. F. Li, Q. H. Zhang, H. Zou, Block-sparse recovery via redundant block OMP, Signal Process., 97 (2014), 162-171. doi: 10.1016/j.sigpro.2013.10.030.

[28]

B. X. Huang, T. Zhou, Recovery of block sparse signals by a block version of StOMP, Signal Process., 106 (2015), 231-244. doi: 10.1016/j.sigpro.2014.07.023.

[29]

J. Huang, T. Zhang, The benefit of group sparsity, Ann. Stat., 38 (2010), 1978-2004. doi: 10.1214/09-AOS778.

[30]

J. H. Lin, S. Li, Block sparse recovery via mixed $\ell_2/\ell_1 $ minimization, Acta Math. Sin., 29 (2013), 1401-1412. doi: 10.1007/s10114-013-1564-y.

[31]

E. Liu, V. N. Temlyakov, The orthogonal super greedy algorithm and applications in compressed sensing, IEEE Trans. Inf. Theory, 58 (2012), 2040-2047. doi: 10.1109/TIT.2011.2177632.

[32]

M. Lustig, D. L. Donoho, J. M. Santos, J. M. Pauly, Compressed sensing MRI, IEEE Signal Process. Mag., 27 (2008), 72-82.

[33]

A. Majumdar, R. K. Ward, Compressed sensing of color images, Signal Process., 90 (2010), 3122-3127. doi: 10.1016/j.sigpro.2010.05.016.

[34]

M. Mishali, Y. C. Eldar, Blind multi-band signal reconstruction: Compressed sensing for analog signals, IEEE Trans. Signal Process., 57 (2009), 993-1009. doi: 10.1109/TSP.2009.2012791.

[35]

M. Mishali, Y. C. Eldar, Reduce and boost: Recovering arbitrary sets of jointly sparse vectors, IEEE Trans. Signal Process., 56 (2008), 4692-4702. doi: 10.1109/TSP.2008.927802.

[36]

Q. Mo, Y. Shen, A remark on the restricted isometry property in orthogonal matching pursuit, IEEE Trans. Inf. Theory, 58 (2012), 3654-3656. doi: 10.1109/TIT.2012.2185923.

[37]

Q. Mo, A sharp restricted isometry constant bound of orthogonal matching pursuit, arXiv: 1501.01708.

[38]

D. Needell, J. A. Troop, CoSaMP: Itertive signal recovery from incomplete and inaccurate samples, Appl. Comput. Harmon. Anal., 26 (2009), 301-321. doi: 10.1016/j.acha.2008.07.002.

[39]

D. Needell, R. Vershynin, Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit, IEEE J. Sel. Top. Signal Process., 4 (2010), 310-316. doi: 10.1109/JSTSP.2010.2042412.

[40]

F. Parvaresh, H. Vikalo, S. Misra, B. Hassibi, Recovering sparse signals using sparse measurement matrices incompressed DNA microarrays, IEEE J. Sel. Top. Signal Process., 2 (2008), 275-285.

[41]

B. D. Rao, K. Kreutz-Delgado, An affine scaling methodology for best basis selection, IEEE Trans. Signal Process., 47 (1999), 187-200. doi: 10.1109/78.738251.

[42]

S. Satpathi, R. L. Das, M. Chakraborty, Improving the bound on the RIP constant in generalized orthogonal matching pursuit, IEEE Signal Proc. Lett., 20 (2013), 1074-1077. doi: 10.1109/LSP.2013.2279977.

[43]

Y. Shen, S. Li, Sparse signals recovery from noisy measurements by orthogonal matching pursuit, Inverse Problems and Imaging, 9 (2015), 231-238. doi: 10.3934/ipi.2015.9.231.

[44]

G. Swirszcz, N. Abe, A. C. Lozano, Grouped orthogonal matching pursuit for variable selection and prediction, Advances in Neural Information Processing Systems, (2009), 1150-1158.

[45]

J. A. Tropp, Greed is Good: Algorithmic results for sparse approximation, IEEE Trans. Inf. Theory, 50 (2004), 2231-2242. doi: 10.1109/TIT.2004.834793.

[46]

J. A. Tropp, Algorithms for simultaneous sparse approximation. Part Ⅰ: Greedy pursuit, Signal Process., 86 (2006), 572-588. doi: 10.1016/j.sigpro.2005.05.030.

[47]

J. A. Tropp, A. C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inf. Theory, 53 (2007), 4655-4666. doi: 10.1109/TIT.2007.909108.

[48]

J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, R. G. Baraniuk, Beyond Nyquist: Efficient sampling of sparse bandlimited signals, IEEE Trans. Inf. Theory, 56 (2010), 520-544. doi: 10.1109/TIT.2009.2034811.

[49]

J. Wang, B. Shim, On the recovery limit of sparse signals using orthogonal matching pursuit, IEEE Trans. Signal Process., 60 (2012), 4973-4976. doi: 10.1109/TSP.2012.2203124.

[50]

J. Wang, S. Kwon, B. Shim, Generalized orthogonal matching pursuit, IEEE Trans. Signal Process., 60 (2012), 6202-6216. doi: 10.1109/TSP.2012.2218810.

[51]

Y. Wang, J. J. Wang, Z. B. Xu, Restricted $p$-isometry properties of nonconvex block-sparse compressed sensing, Signal Process., 104 (2014), 188-196. doi: 10.1016/j.sigpro.2014.03.040.

[52]

Y. Wang, J. J. Wang, Z. B. Xu, On recovery of block-sparse signals via mixed $\ell_2/\ell_p(0 < p≤q1) $ norm minimization, EURASIP J. Adv. Signal Process., 76 (2013), 1-17.

[53]

J. M. Wen, Z. C. Zhou, Z. L. Liu, M. J. Lai and X. H. Tang, Sharp sufficient conditions for stable recovery of block sparse signals by block orthogonal matching pursuit, arXiv: 1605.02894.

[54]

J. M. Wen, Z. C. Zhou, D. F. Li, X. H. Tang, A novel sufficient condition for generalized orthogonal matching pursuit, IEEE Communications Letters, 21 (2017), 805-808. doi: 10.1109/LCOMM.2016.2642922.

[55]

J. M. Wen, Z. C. Zhou, J. Wang, X. H. Tang, Q. Mo, A sharp condition for exact support recovery with orthogonal matching pursuit, IEEE Trans. Signal Process., 65 (2017), 1370-1382. doi: 10.1109/TSP.2016.2634550.

[56]

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Anal. Mach. Intell., 31 (2009), 210-227. doi: 10.1109/AFGR.2008.4813404.

[57]

R. Wu, W. Huang, D. R. Chen, The exact support recovery of sparse signals with noise via orthogonal matching pursuit, IEEE Signal Proc. Let., 20 (2013), 403-406.

[58]

Y. Xu, X. H. Qiu, Block-sparse signals recovery using orthogonal multimatching (Chinese), Journal of Signal Processing, 30 (2014), 706-711.

[59]

Z. Q. Xu, The performance of orthogonal multi-matching pursuit under RIP, J. Comp. Math. Sci., 33 (2015), 495-516. doi: 10.4208/jcm.1505-m4529.

[60]

Q. Zhao, J. K. Wang, Y. H. Han, P. Han, Compressive sensing of block-sparse signals recovery based on sparsity adaptive regularized orthogonal matching pursuit algorithm, IEEE Fifth International Conference on Advanced Computational Intelligence, (2012), 1141-1144.

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