# zbMATH — the first resource for mathematics

Fast thresholding algorithms with feedbacks and partially known support for compressed sensing. (English) Zbl 1457.94037
Summary: Some works in modified compressive sensing (CS) show that reconstruction of sparse signals can obtain better results than traditional CS using the partially known support. In this paper, we extend the idea of these works to the null space tuning algorithm with hard thresholding, feedbacks (NST + HT + FB) and derive sufficient conditions for robust sparse signal recovery. The theoretical analysis shows that including prior information of partially known support relaxes the preconditioned restricted isometry property condition comparing with the NST + HT + FB. Numerical experiments demonstrate that the modification improves the performance of the NST+HT+FB, thereby requiring fewer samples to obtain an approximate reconstruction. Meanwhile, a systemic comparison with different methods based on partially known support is shown.
Reviewer: Reviewer (Berlin)
##### MSC:
 94A12 Signal theory (characterization, reconstruction, filtering, etc.)
CoSaMP; PDCO
Full Text:
##### References:
 [1] Baraniuk, RG, Cevher, V, Duarte, MF and Hegde, C (2010). Model-based compressive sensing. IEEE Transactions on Information Theory, 56(4), 1982-2001. · Zbl 1366.94215 [2] Bredies, K and Lorenz, DA (2011). Iterated hard shrinkage for minimization problems with sparsity constraints. Siam Journal on Scientific Computing, 30(2), 657-683. · Zbl 1170.46067 [3] Cai, T, Xu, G and Zhang, J (2009). On recovery of sparse signals via $$l_1$$ minimization. IEEE Transactions on Information Theory, 55(7), 3388-3397. · Zbl 1367.94081 [4] Candès, EJ and Tao, T (2005). Decoding by linear programming. IEEE Transactions on Information Theory, 51(12), 4203-4215. · Zbl 1264.94121 [5] Carrillo, RE, Polania, LF and Barner, KE (2011). Iterative hard thresholding for compressed sensing with partially known support. In IEEE International Conference on Acoustics Speech and Signal Processing, pp. 4028-4031. [6] Carrillo, RE, Polania, LF and Barner, KE (2010). Iterative algorithms for compressed sensing with partially known support. In IEEE International Conference on Acoustics Speech and Signal Processing, pp. 3654-3657. [7] Chen, SS, Donoho, DL and Saunders, MA (2001). Atomic decomposition by basis pursuit. Siam Review, 43(1), 129-159. · Zbl 0979.94010 [8] Dai, W and Milenkovic, O (2009). Subspace pursuit for compressive sensing signal reconstruction. IEEE Transactions on Information Theory, 55(5), 2230-2249. · Zbl 1367.94082 [9] Donoho, DL (2006). Compressed sensing. IEEE Transactions on Information Theory, 52(4), 1289-1306. · Zbl 1288.94016 [10] Duarte, MF, Hegde, C, Cevher, V and Baraniuk, RG (2009). Recovery of compressible signals in unions of subspaces. In Information Sciences and Systems, pp. 175-180. [11] Figueiredo, MAT, Nowak, RD and Wright, SJ (2008). Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems. IEEE Journal of Selected Topics in Signal Processing, 1(4), 586-597. [12] Foucart, S (2011). Hard thresholding pursuit: An algorithm for compressive sensing. Siam Journal on Numerical Analysis, 49(6), 2543-2563. · Zbl 1242.65060 [13] Jinseon, L (2009). A short note on compressed sensing with partially known signal support. Signal Processing, 90(12), 3308-3312. [14] Kim, SJ, Koh, K, Lustig, M, Boyd, S and Gorinevsky, D (2008). An interior-point method for large-scale l1-regularized least squares. IEEE Journal of Selected Topics in Signal Processing, 1(4), 606-617. [15] Li, S, Liu, Y and Mi, T (2014). Fast thresholding algorithms with feedbacks for sparse signal recovery. Applied and Computational Harmonic Analysis, 37(1), 69-88. · Zbl 1294.65068 [16] Lin, J and Li, S (2016). Restricted $$q$$-isometry properties adapted to frames for nonconvex $$l_q$$-analysis. IEEE Transactions on Information Theory, 62(8), 4733-4747. · Zbl 1359.94125 [17] Mallat, SG and Zhang, Z (1993). Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41(12), 3397-3415. · Zbl 0842.94004 [18] Mo, Q and Yi, S (2012). A remark on the restricted isometry property in orthogonal matching pursuit. IEEE Transactions on Information Theory, 58(6), 3654-3656. · Zbl 1365.94182 [19] Natarajan, BK (1995). Sparse approximate solutions to linear systems. Siam Journal on Computing, 24(2), 227-234. · Zbl 0827.68054 [20] Needell, D and Tropp, JA (2009). Cosamp: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis, 26(3), 301-321. · Zbl 1163.94003 [21] Needell, D and Vershynin, R (2007). Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Foundations of Computational Mathematics, 9(3), 317-334. · Zbl 1183.68739 [22] Vaswani, N and Lu, W (2010). Modified-cs: Modifying compressive sensing for problems with partially known support. IEEE Transactions on Signal Processing, 58(9), 4595-4607. · Zbl 1392.94045 [23] Wu, R and Huang, W (2013). Hard thresholding pursuit with partially known support for compressed sensing. Advanced Materials Research, 718, 669-674. [24] Zhang, R and Li, S (2018). A proof of conjecture on restricted isometry property constants \(\delta_{t k}(0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.