# American Institute of Mathematical Sciences

August  2020, 19(8): 4055-4068. doi: 10.3934/cpaa.2020179

## Analysis non-sparse recovery for relaxed ALASSO

 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author

Received  August 2019 Revised  November 2019 Published  May 2020

Fund Project: This work was supported in part by key project of NSF of China under grant number 11531013, the fundamental research funds for the central universities, and the NSAF of China under grant number U1630116

This paper considers recovery of signals that are corrupted with noise. We focus on a novel model which is called relaxed ALASSO (RALASSO) model introduced by Z. Tan et al. (2014). Compared to the well-known ALASSO, RALASSO can be solved better in practice. Z. Tan et al. (2014) used the $D$-RIP to characterize the sparse or approximately sparse solutions for RALASSO when the $D$-RIP constant $\delta_{2k} < 0.1907$, where the solution is sparse or approximately sparse in terms of a tight frame $D$. However, their estimate of error bound for solution heavily depends on the term $\Vert D^*D\Vert_{1, 1}$. Besides, compared to other works on signals recovering from ALASSO, the condition $\delta_{2k} < 0.1907$ is even stronger. Based on the RALASSO model, we use new methods to get a better estimate of error bound and give a weaker sufficient condition in this article for the inadequacies of the results by Z. Tan et al. (2014). One of the result of this paper is to use another method called the robust $\ell_2$ $D$-Null Space Property to obtain the sparse or non-sparse solution of RALASSO and give the error estimation of RALASSO, where we eliminate the term $\Vert D^*D\Vert_{1, 1}$ in the constants. Another result of the paper is to utilize the $D$-RIP to obtain a new condition $\delta_{2k} < 0.3162$ which is weaker than the condition $\delta_{2k} < 0.1907$. To some extent, RALASSO is equivalent to ALASSO and the condition is also weaker than the similar one $\delta_{3k} < 0.25$ by J. Lin, and S. Li (2014) and $\delta_{2k}<0.25$ by Y. Xia, and S. Li (2016).

Citation: Hang Xu, Song Li. Analysis non-sparse recovery for relaxed ALASSO. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4055-4068. doi: 10.3934/cpaa.2020179
##### References:
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##### References:
 [1] M. V. Afonso, J. M. Bioucas-Dias and and M. A. T. Figueiredo, Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 9 (2010), 2345-2356.  doi: 10.1109/TIP.2010.2047910. [2] A. Beck and and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Imag. Sci., 2 (2009), 183-202.  doi: 10.1137/080716542. [3] A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process, 18 (2009), 2419-2434.  doi: 10.1109/TIP.2009.2028250. [4] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, Cambridge, U.K., 2004.  doi: 10.1017/CBO9780511804441. [5] S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2010), 1-122.  doi: 10.1561/2200000016. [6] T. Cai and A. Zhang, Sparse representation of a polytope and recovery of sparse signals and low-rank matrices, IEEE Trans. Inform. Theory, 60 (2014), 122-132.  doi: 10.1109/TIT.2013.2288639. [7] E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), 489-509.  doi: 10.1109/TIT.2005.862083. [8] E. J. Candès, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59 (2006), 1207-1223.  doi: 10.1002/cpa.20124. [9] E. J. Candès, Y. C. Eldar, D. Needell and P. Randall, Compressed sensing with coherent and redundant dictionaries, Appl. Comput. Harmon. Anal., 31 (2011), 59-73.  doi: 10.1016/j.acha.2010.10.002. [10] S. S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM Rev., 43 (2001), 129-159.  doi: 10.1137/S003614450037906X. [11] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306.  doi: 10.1109/TIT.2006.871582. [12] D. L. Donoho, Y. Tsaig, I. Drori and J. L. Starck, Sparse solution of underdetermined linear equations by stagewise Orthogonal Matching Pursuit, IEEE Trans. Inform. Theory, 58 (2012), 1094-1121.  doi: 10.1109/TIT.2011.2173241. [13] I. Drori, Fast $\ell_1$-minimization by iterative thresholding for multidimensional NMR spectroscopy, EURASIP J. Adv. Signal Process., 2007 (2007), 1-10.  doi: 10.1155/2007/20248. [14] M. Elad, P. Milanfar and R. Rubinstein, Analysis versus synthesis in signal priors, Inverse Probl., 23 (2007), 947-968.  doi: 10.1088/0266-5611/23/3/007. [15] S. Fourcart, Stability and robustness of $\ell_1$-minimization with Weibull matrices and redundant dictionaries, Linear Alg. Appl., 441 (2014), 4-21.  doi: 10.1016/j.laa.2012.10.003. [16] M. Herman and T. Strohmer, High-resolution radar via compressed sensing, IEEE Trans. Signal Process, 57 (2009), 2275-2284.  doi: 10.1109/TSP.2009.2014277. [17] J. Lin and S. Li, Sparse recovery with coherent tight frame via analysis Dantzig selector and analysis lasso, Appl. Comput. Harmon. Anal., 37 (2014), 126-139.  doi: 10.1016/j.acha.2013.10.003. [18] M. Lustig, D. L. Donoho, J. M. Santos and J. M. Pauly, Compressed sensing MRI, IEEE Signal Process Mag., 27 (2008), 72-82.  doi: 10.1109/TIT.2006.871582. [19] S. Nam, M. E. Davies, M. Elad and R. Gribonval, The cosparse analysis model and algorithms, Appl. Comput. Harmon. Anal., 34 (2013), 30-56.  doi: 10.1016/j.acha.2012.03.006. [20] B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput., 24 (1995), 227-234.  doi: 10.1137/S0097539792240406. [21] D. Needell and and J. A. Tropp, Cosamp: Iterative signal recovery from noisy samples, Appl. Comput. Harmon. Anal., 26 (2008), 301-321.  doi: 10.1016/j.acha.2008.07.002. [22] H. Rauhut, K. Schnass and P. Vandergheynst, Compressed sensing and redundant dictionaries, IEEE Trans. Inform. Theory, 54 (2008), 2210-2219.  doi: 10.1109/TIT.2008.920190. [23] Z. Tan, Y. C. Eldar, A. Beck and A. Nehorai, Smoothing and decomposition for analysis sparse recovery, IEEE Trans. Signal Process, 62 (2014), 1762-1774.  doi: 10.1109/TSP.2014.2304932. [24] R. Tibshirani, Regression shrinkage and selection via lasso, J. R. Statist. Soc. B. Stat. Meth., 58 (1996), 267-288. [25] R. Tibshirani, M. Saunders, S. Rosset, J. Zhu and K. Knight, Sparsity and smoothness via the fused Lasso, J. R. Statist. Soc. B. Stat. Meth., 67 (2005), 91-108.  doi: 10.1111/j.1467-9868.2005.00490.x. [26] Y. Xia and S. Li, Analysis recovery with coherent frames and correlated measurements, IEEE Trans. Inform. Theory, 62 (2016), 6493-6507.  doi: 10.1109/TIT.2016.2606638. [27] R. Zhang and S. Li, Optimal RIP bounds for sparse signals recovery via $\ell_p$ minimization, Appl. Comput. Harmon. Anal., 47 (2019), 566-584.  doi: 10.1016/j.acha.2017.10.004.
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