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February  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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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 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. [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. [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. [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. [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 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. [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. [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. [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 and 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 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. [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. [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. [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. [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. [29] J. Huang and T. Zhang, The benefit of group sparsity, Ann. Stat., 38 (2010), 1978-2004.  doi: 10.1214/09-AOS778. [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. [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. [32] M. Lustig, D. L. Donoho, J. M. Santos and J. M. Pauly, Compressed sensing MRI, IEEE Signal Process. Mag., 27 (2008), 72-82. [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. [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. [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. [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. [37] Q. Mo, A sharp restricted isometry constant bound of orthogonal matching pursuit, arXiv: 1501.01708. [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. [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. [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. [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. [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. [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. [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. [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 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. [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. [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. [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. [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. [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. [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 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. [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. [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. [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. [58] Y. Xu and 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 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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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 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. [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. [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. [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. [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 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. [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. [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. [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 and 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 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. [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. [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. [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. [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. [29] J. Huang and T. Zhang, The benefit of group sparsity, Ann. Stat., 38 (2010), 1978-2004.  doi: 10.1214/09-AOS778. [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. [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. [32] M. Lustig, D. L. Donoho, J. M. Santos and J. M. Pauly, Compressed sensing MRI, IEEE Signal Process. Mag., 27 (2008), 72-82. [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. [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. [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. [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. [37] Q. Mo, A sharp restricted isometry constant bound of orthogonal matching pursuit, arXiv: 1501.01708. [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. [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. [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. [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. [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. [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. [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. [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 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. [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. [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. [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. [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. [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. [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 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. [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. [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. [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. [58] Y. Xu and 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 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.
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