Startseite Mathematik Recovery analysis for the ℓ p /ℓ1 minimization problem
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Recovery analysis for the ℓ p /ℓ1 minimization problem

  • Zhiyong Zhou ORCID logo EMAIL logo
Veröffentlicht/Copyright: 15. November 2024

Abstract

In this paper, we present a theoretical analysis of the p / 1 minimization method with 0 < p < 1 for sparse signal recovery. We provide a verifiable sufficient condition for the exact noiseless sparse recovery and establish reconstruction error bounds using q-ratio constrained minimal singular values (CMSV) and restricted isometry property (RIP) tools. Additionally, we adopt an efficient algorithm to solve the optimization problem and conduct numerical experiments to demonstrate its superior performance.

MSC 2020: 94A12; 94A20

Award Identifier / Grant number: 12201556

Award Identifier / Grant number: 12361054

Funding statement: This work was supported by the National Natural Science Foundation of China (Grants No. 12201556 and No. 12361054), and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ21A010003).

References

[1] R. I. Boţ, M. N. Dao and G. Li, Extrapolated proximal subgradient algorithms for nonconvex and nonsmooth fractional programs, Math. Oper. Res. 47 (2022), no. 3, 2415–2443. 10.1287/moor.2021.1214Suche in Google Scholar

[2] S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn. 3 (2011), no. 1, 1–122. Suche in Google Scholar

[3] T. T. Cai and A. Zhang, Sharp RIP bound for sparse signal and low-rank matrix recovery, Appl. Comput. Harmon. Anal. 35 (2013), no. 1, 74–93. 10.1016/j.acha.2012.07.010Suche in Google Scholar

[4] T. T. Cai and A. Zhang, Sparse representation of a polytope and recovery in sparse signals and low-rank matrices, IEEE Trans. Inform. Theory 60 (2014), no. 1, 122–132. 10.1109/TIT.2013.2288639Suche in Google Scholar

[5] E. J. Candes, The restricted isometry property and its implications for compressed sensing, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9–10, 589–592. 10.1016/j.crma.2008.03.014Suche in Google Scholar

[6] E. J. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory 51 (2005), no. 12, 4203–4215. 10.1109/TIT.2005.858979Suche in Google Scholar

[7] R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, IEEE Press, Piscataway (2008), 3869–3872. 10.1109/ICASSP.2008.4518498Suche in Google Scholar

[8] A. Cohen, W. Dahmen and R. DeVore, Compressed sensing and best k-term approximation, J. Amer. Math. Soc. 22 (2009), no. 1, 211–231. 10.1090/S0894-0347-08-00610-3Suche in Google Scholar

[9] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289–1306. 10.1109/TIT.2006.871582Suche in Google Scholar

[10] Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications, Cambridge University Press, Cambridge, 2012. 10.1017/CBO9780511794308Suche in Google Scholar

[11] J. Fan and R. Li, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc. 96 (2001), no. 456, 1348–1360. 10.1198/016214501753382273Suche in Google Scholar

[12] S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2013. 10.1007/978-0-8176-4948-7Suche in Google Scholar

[13] S. Huang and T. D. Tran, Sparse signal recovery via generalized entropy functions minimization, IEEE Trans. Signal Process. 67 (2019), no. 5, 1322–1337. 10.1109/TSP.2018.2889951Suche in Google Scholar

[14] H. Li and Z. Lin, Accelerated proximal gradient methods for nonconvex programming, Advances in Neural Information Processing Systems 28, Curran Associates, Red Hook (2015), 379–387. Suche in Google Scholar

[15] Q. Li, L. Shen, N. Zhang and J. Zhou, A proximal algorithm with backtracked extrapolation for a class of structured fractional programming, Appl. Comput. Harmon. Anal. 56 (2022), 98–122. 10.1016/j.acha.2021.08.004Suche in Google Scholar

[16] M. E. Lopes, Unknown sparsity in compressed sensing: Denoising and inference, IEEE Trans. Inform. Theory 62 (2016), no. 9, 5145–5166. 10.1109/TIT.2016.2587772Suche in Google Scholar

[17] T.-H. Ma, Y. Lou and T.-Z. Huang, Truncated l 1 - 2 models for sparse recovery and rank minimization, SIAM J. Imaging Sci. 10 (2017), no. 3, 1346–1380. 10.1137/16M1098929Suche in Google Scholar

[18] C. Moler, Generate figures for Cleve’s corner on compressed sensing, (2016), https://ww2.mathworks.cn/matlabcentral/fileexchange/28250-generate-figures-for-cleve-s-corner-on-compressed-sensing. Suche in Google Scholar

[19] V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, New York, 1984. 10.1007/978-1-4612-5280-1Suche in Google Scholar

[20] Y. Rahimi, C. Wang, H. Dong and Y. Lou, A scale-invariant approach for sparse signal recovery, SIAM J. Sci. Comput. 41 (2019), no. 6, A3649–A3672. 10.1137/18M123147XSuche in Google Scholar

[21] M. Tao, Minimization of L 1 over L 2 for sparse signal recovery with convergence guarantee, SIAM J. Sci. Comput. 44 (2022), no. 2, A770–A797. 10.1137/20M136801XSuche in Google Scholar

[22] R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B 58 (1996), no. 1, 267–288. 10.1111/j.2517-6161.1996.tb02080.xSuche in Google Scholar

[23] A. N. Tichonov, A. S. Leonov and A. G. Yagola, Nonlinear Ill-Posed Problems. Vol. 1, Chapman & Hall, London, 1998. 10.1007/978-94-017-5167-4_1Suche in Google Scholar

[24] C. Wang, J.-F. Aujol, G. Gilboa and Y. Lou, Minimizing quotient regularization model, preprint (2023), https://arxiv.org/abs/2308.04095. Suche in Google Scholar

[25] C. Wang, M. Tao, C.-N. Chuah, J. Nagy and Y. Lou, Minimizing L 1 over L 2 norms on the gradient, Inverse Problems 38 (2022), no. 6, Article ID 065011. 10.1088/1361-6420/ac64fbSuche in Google Scholar

[26] C. Wang, M. Yan, Y. Rahimi and Y. Lou, Accelerated schemes for the L 1 / L 2 minimization, IEEE Trans. Signal Process. 68 (2020), 2660–2669. 10.1109/TSP.2020.2985298Suche in Google Scholar

[27] J. Wang and Q. Ma, The variant of the iterative shrinkage-thresholding algorithm for minimization of the 1 over norms, Signal Process. 211 (2023), Article ID 109104. 10.1016/j.sigpro.2023.109104Suche in Google Scholar

[28] F. Wen, L. Chu, P. Liu and R. C. Qiu, A survey on nonconvex regularization-based sparse and low-rank recovery in signal processing, statistics, and machine learning, IEEE Access 6 (2018), 69883–69906. 10.1109/ACCESS.2018.2880454Suche in Google Scholar

[29] Y. Xie, X. Su and H. Ge, RIP analysis for 1 / p ( p > 1 ) minimization method, IEEE Signal Process. Lett. 30 (2023), 997–1001. 10.1109/LSP.2023.3298283Suche in Google Scholar

[30] Y. Xu, A. Narayan, H. Tran and C. G. Webster, Analysis of the ratio of 1 and 2 norms in compressed sensing, Appl. Comput. Harmon. Anal. 55 (2021), 486–511. 10.1016/j.acha.2021.06.006Suche in Google Scholar

[31] P. Yin, Y. Lou, Q. He and J. Xin, Minimization of 1 - 2 for compressed sensing, SIAM J. Sci. Comput. 37 (2015), no. 1, A536–A563. 10.1137/140952363Suche in Google Scholar

[32] L. Zeng, P. Yu and T. K. Pong, Analysis and algorithms for some compressed sensing models based on L1/L2 minimization, SIAM J. Optim. 31 (2021), no. 2, 1576–1603. 10.1137/20M1355380Suche in Google Scholar

[33] C.-H. Zhang, Nearly unbiased variable selection under minimax concave penalty, Ann. Statist. 38 (2010), no. 2, 894–942. 10.1214/09-AOS729Suche in Google Scholar

[34] R. Zhang and S. Li, A proof of conjecture on restricted isometry property constants δ t k ( 0 < t < 4 3 ) , IEEE Trans. Inform. Theory 64 (2018), no. 3, 1699–1705. 10.1109/TIT.2017.2705741Suche in Google Scholar

[35] R. Zhang and S. Li, Optimal RIP bounds for sparse signals recovery via p minimization, Appl. Comput. Harmon. Anal. 47 (2019), no. 3, 566–584. 10.1016/j.acha.2017.10.004Suche in Google Scholar

[36] S. Zhang and J. Xin, Minimization of transformed L 1 penalty: Theory, difference of convex function algorithm, and robust application in compressed sensing, Math. Program. 169 (2018), no. 1, 307–336. 10.1007/s10107-018-1236-xSuche in Google Scholar

[37] Z. Zhou, A unified framework for constructing nonconvex regularizations, IEEE Signal Process. Lett. 29 (2022), 479–483. 10.1109/LSP.2022.3140709Suche in Google Scholar

[38] Z. Zhou, RIP analysis for the weighted r - 1 minimization method, Signal Process. 202 (2023), Article ID 108754. 10.1016/j.sigpro.2022.108754Suche in Google Scholar

[39] Z. Zhou and J. Yu, Sparse recovery based on q-ratio constrained minimal singular values, Signal Process. 155 (2019), 247–258. 10.1016/j.sigpro.2018.10.002Suche in Google Scholar

[40] Z. Zhou and J. Yu, Minimization of the q-ratio sparsity with 1 < q for signal recovery, Signal Process. 189 (2021), Article ID 108250. 10.1016/j.sigpro.2021.108250Suche in Google Scholar

Received: 2024-07-10
Revised: 2024-10-12
Accepted: 2024-10-28
Published Online: 2024-11-15
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 10.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jiip-2024-0044/pdf
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