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January  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 and P. Steeghs, Compressive radar imaging, in Proc. IEEE Radar Conf., 303 (2007), 128-133. doi: 10.1109/RADAR.2007.374203. Google Scholar [2] T. Blumensath and M. E. Davies, Iterative thresholding for sparse approximations, J. Fourier Anal. Appl., 14 (2008), 629-654. doi: 10.1007/s00041-008-9035-z. Google Scholar [3] T. Blumensath and 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. Google Scholar [4] T. T. Cai and 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. Google Scholar [5] T. T. Cai and 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. Google Scholar [6] E. J. Candés, M. B. Wakin and 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. Google Scholar [7] E. J. Candés, J. Romberg and 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. Google Scholar [8] E. J. Candés and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979. Google Scholar [9] J. Chen and 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. Google Scholar [10] S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), 33-61. doi: 10.1137/S1064827596304010. Google Scholar [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. Google Scholar [12] W. G. Chen and 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. Google Scholar [13] S. F. Cotter, B. D. Rao, K. Engan and 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. Google Scholar [14] W. Dai and O. Milenkovic, Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory, 55 (2009), 2230-2249. doi: 10.1109/TIT.2009.2016006. Google Scholar [15] W. Dan and 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. Google Scholar [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. Google Scholar [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. Google Scholar [18] I. Daubechies, M. Defrise and 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. Google Scholar [19] M. A. Davenport and 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. Google Scholar [20] D. L. Donoho, I. Drori, Y. Tsaig and 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. Google Scholar [21] D. L. Donoho, Denoising by soft-threshold, IEEE Trans. Inf. Theory, 41 (1995), 613-627. doi: 10.1109/18.382009. Google Scholar [22] D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. Google Scholar [23] D. L. Donoho and X. Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theory, 47 (2001), 2845-2862. doi: 10.1109/18.959265. Google Scholar [24] Y. C. Eldar, P. Kuppinger and H. Bölcskei, Block-sparse signals: uncertainty relations and efficient recovery, IEEE Trans. Signal Process., 58 (2010), 3042-3054. doi: 10.1109/TSP.2010.2044837. Google Scholar [25] Y. C. Eldar and 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. Google Scholar [26] Y. C. Eldar and 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. Google Scholar [27] Y. L. Fu, H. F. Li, Q. H. Zhang and H. Zou, Block-sparse recovery via redundant block OMP, Signal Process., 97 (2014), 162-171. doi: 10.1016/j.sigpro.2013.10.030. Google Scholar [28] B. X. Huang and 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. Google Scholar [29] J. Huang and T. Zhang, The benefit of group sparsity, Ann. Stat., 38 (2010), 1978-2004. doi: 10.1214/09-AOS778. Google Scholar [30] J. H. Lin and 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. Google Scholar [31] E. Liu and 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. Google Scholar [32] M. Lustig, D. L. Donoho, J. M. Santos and J. M. Pauly, Compressed sensing MRI, IEEE Signal Process. Mag., 27 (2008), 72-82. Google Scholar [33] A. Majumdar and R. K. Ward, Compressed sensing of color images, Signal Process., 90 (2010), 3122-3127. doi: 10.1016/j.sigpro.2010.05.016. Google Scholar [34] M. Mishali and 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. Google Scholar [35] M. Mishali and 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. Google Scholar [36] Q. Mo and 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. Google Scholar [37] Q. Mo, A sharp restricted isometry constant bound of orthogonal matching pursuit, arXiv: 1501.01708.Google Scholar [38] D. Needell and 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. Google Scholar [39] D. Needell and 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. Google Scholar [40] F. Parvaresh, H. Vikalo, S. Misra and B. Hassibi, Recovering sparse signals using sparse measurement matrices incompressed DNA microarrays, IEEE J. Sel. Top. Signal Process., 2 (2008), 275-285. Google Scholar [41] B. D. Rao and K. Kreutz-Delgado, An affine scaling methodology for best basis selection, IEEE Trans. Signal Process., 47 (1999), 187-200. doi: 10.1109/78.738251. Google Scholar [42] S. Satpathi, R. L. Das and 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. Google Scholar [43] Y. Shen and 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. Google Scholar [44] G. Swirszcz, N. Abe and A. C. Lozano, Grouped orthogonal matching pursuit for variable selection and prediction, Advances in Neural Information Processing Systems, (2009), 1150-1158. Google Scholar [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. Google Scholar [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. Google Scholar [47] J. A. Tropp and 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. Google Scholar [48] J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg and 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. Google Scholar [49] J. Wang and 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. Google Scholar [50] J. Wang, S. Kwon and B. Shim, Generalized orthogonal matching pursuit, IEEE Trans. Signal Process., 60 (2012), 6202-6216. doi: 10.1109/TSP.2012.2218810. Google Scholar [51] Y. Wang, J. J. Wang and 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. Google Scholar [52] Y. Wang, J. J. Wang and 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. Google Scholar [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.Google Scholar [54] J. M. Wen, Z. C. Zhou, D. F. Li and 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. Google Scholar [55] J. M. Wen, Z. C. Zhou, J. Wang, X. H. Tang and 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. Google Scholar [56] J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry and Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Anal. Mach. Intell., 31 (2009), 210-227. doi: 10.1109/AFGR.2008.4813404. Google Scholar [57] R. Wu, W. Huang and D. R. Chen, The exact support recovery of sparse signals with noise via orthogonal matching pursuit, IEEE Signal Proc. Let., 20 (2013), 403-406. Google Scholar [58] Y. Xu and X. H. Qiu, Block-sparse signals recovery using orthogonal multimatching (Chinese), Journal of Signal Processing, 30 (2014), 706-711. Google Scholar [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. Google Scholar [60] Q. Zhao, J. K. Wang, Y. H. Han and 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. Google Scholar

show all references

##### References:
 [1] R. Baraniuk and P. Steeghs, Compressive radar imaging, in Proc. IEEE Radar Conf., 303 (2007), 128-133. doi: 10.1109/RADAR.2007.374203. Google Scholar [2] T. Blumensath and M. E. Davies, Iterative thresholding for sparse approximations, J. Fourier Anal. Appl., 14 (2008), 629-654. doi: 10.1007/s00041-008-9035-z. Google Scholar [3] T. Blumensath and 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. Google Scholar [4] T. T. Cai and 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. Google Scholar [5] T. T. Cai and 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. Google Scholar [6] E. J. Candés, M. B. Wakin and 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. Google Scholar [7] E. J. Candés, J. Romberg and 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. Google Scholar [8] E. J. Candés and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979. Google Scholar [9] J. Chen and 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. Google Scholar [10] S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), 33-61. doi: 10.1137/S1064827596304010. Google Scholar [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. Google Scholar [12] W. G. Chen and 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. Google Scholar [13] S. F. Cotter, B. D. Rao, K. Engan and 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. Google Scholar [14] W. Dai and O. Milenkovic, Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory, 55 (2009), 2230-2249. doi: 10.1109/TIT.2009.2016006. Google Scholar [15] W. Dan and 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. Google Scholar [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. Google Scholar [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. Google Scholar [18] I. Daubechies, M. Defrise and 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. Google Scholar [19] M. A. Davenport and 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. Google Scholar [20] D. L. Donoho, I. Drori, Y. Tsaig and 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. Google Scholar [21] D. L. Donoho, Denoising by soft-threshold, IEEE Trans. Inf. Theory, 41 (1995), 613-627. doi: 10.1109/18.382009. Google Scholar [22] D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. Google Scholar [23] D. L. Donoho and X. Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theory, 47 (2001), 2845-2862. doi: 10.1109/18.959265. Google Scholar [24] Y. C. Eldar, P. Kuppinger and H. Bölcskei, Block-sparse signals: uncertainty relations and efficient recovery, IEEE Trans. Signal Process., 58 (2010), 3042-3054. doi: 10.1109/TSP.2010.2044837. Google Scholar [25] Y. C. Eldar and 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. Google Scholar [26] Y. C. Eldar and 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. Google Scholar [27] Y. L. Fu, H. F. Li, Q. H. Zhang and H. Zou, Block-sparse recovery via redundant block OMP, Signal Process., 97 (2014), 162-171. doi: 10.1016/j.sigpro.2013.10.030. Google Scholar [28] B. X. Huang and 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. Google Scholar [29] J. Huang and T. Zhang, The benefit of group sparsity, Ann. Stat., 38 (2010), 1978-2004. doi: 10.1214/09-AOS778. Google Scholar [30] J. H. Lin and 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. Google Scholar [31] E. Liu and 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. Google Scholar [32] M. Lustig, D. L. Donoho, J. M. Santos and J. M. Pauly, Compressed sensing MRI, IEEE Signal Process. Mag., 27 (2008), 72-82. Google Scholar [33] A. Majumdar and R. K. Ward, Compressed sensing of color images, Signal Process., 90 (2010), 3122-3127. doi: 10.1016/j.sigpro.2010.05.016. Google Scholar [34] M. Mishali and 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. Google Scholar [35] M. Mishali and 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. Google Scholar [36] Q. Mo and 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. Google Scholar [37] Q. Mo, A sharp restricted isometry constant bound of orthogonal matching pursuit, arXiv: 1501.01708.Google Scholar [38] D. Needell and 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. Google Scholar [39] D. Needell and 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. Google Scholar [40] F. Parvaresh, H. Vikalo, S. Misra and B. Hassibi, Recovering sparse signals using sparse measurement matrices incompressed DNA microarrays, IEEE J. Sel. Top. Signal Process., 2 (2008), 275-285. Google Scholar [41] B. D. Rao and K. Kreutz-Delgado, An affine scaling methodology for best basis selection, IEEE Trans. Signal Process., 47 (1999), 187-200. doi: 10.1109/78.738251. Google Scholar [42] S. Satpathi, R. L. Das and 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. Google Scholar [43] Y. Shen and 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. Google Scholar [44] G. Swirszcz, N. Abe and A. C. Lozano, Grouped orthogonal matching pursuit for variable selection and prediction, Advances in Neural Information Processing Systems, (2009), 1150-1158. Google Scholar [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. Google Scholar [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. Google Scholar [47] J. A. Tropp and 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. Google Scholar [48] J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg and 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. Google Scholar [49] J. Wang and 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. Google Scholar [50] J. Wang, S. Kwon and B. Shim, Generalized orthogonal matching pursuit, IEEE Trans. Signal Process., 60 (2012), 6202-6216. doi: 10.1109/TSP.2012.2218810. Google Scholar [51] Y. Wang, J. J. Wang and 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. Google Scholar [52] Y. 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