# American Institute of Mathematical Sciences

October  2019, 15(4): 2009-2021. doi: 10.3934/jimo.2018134

## An efficient algorithm for non-convex sparse optimization

 1 School of Mathematics, Tianjin University, Tianjin 300072, China 2 Department of Computing, Curtin University, WA, 6102, Australia 3 Department of Mathematics and Statistics, Curtin University, WA, 6102, Australia

* Corresponding author: Wanquan Liu

Received  March 2017 Revised  April 2018 Published  September 2018

It is a popular research topic in computer vision community to find a solution for the zero norm minimization problem via solving its non-convex relaxation problem. In fact, there are already many existing algorithms to solve the non-convex relaxation problem. However, most of them are computationally expensive due to the non-Lipschitz property of this problem and thus these existing algorithms are not suitable for many engineering problems with large dimensions.

In this paper, we first develop an efficient algorithm to solve the non-convex relaxation problem via solving a sequence of non-convex sub-problems based on our recent work. To this end, we reformulate the minimization problem into another non-convex one but with non-negative constraint. Then we can transform the non-Lipschitz continuous non-convex problem with the non-negative constraint into a Lipschitz continuous problem, which allows us to use some efficient existing algorithms for its solution. Based on the proposed algorithm, an important relation between the solutions of relaxation problem and the original zero norm minimization problem is established from a different point of view. The results in this paper reveal two important issues: ⅰ) The solution of non-convex relaxation minimization problem converges to the solution of the original problem; ⅱ) The general non-convex relaxation problem can be solved efficiently with another reformulated high dimension problem with nonnegative constraint. Finally, some numerical results are used to demonstrate effectiveness of the proposed algorithm.

Citation: Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134
##### References:
 [1] R. H. Byrd, P. Lu and J. Nocedal, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Stat. Comput., 16 (1995), 1190-1208. doi: 10.1137/0916069. 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. Learning, 3 (2010), 1-122. Google Scholar [3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441. Google Scholar [4] A. Cohen, W. Dahmen and R. DeVore, Compressed sensing and best $k$-term approximation, J. Amer. Math. Soc., 22 (2009), 211-231. doi: 10.1090/S0894-0347-08-00610-3. Google Scholar [5] R. Chartrand, Nonconvex compressed sensing and error correction, IEEE International Conference on Acoustics, Speech and Signal Processing, (2007), 889-892. Google Scholar [6] A. Charkrabarti and F. 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Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979. Google Scholar [12] X. Chen and S. Xiang, Sparse solutions of linear complementarity problems, Math. Program., 159 (2016), 539-556. doi: 10.1007/s10107-015-0950-x. Google Scholar [13] R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing, IEEE International Conference on Acoustics, Speech and Signal Processing, (2008), 3869-3872. Google Scholar [14] X. Chen and W. Zhou, Convergence of Reweighted $l_1$ Minimization Algorithms and Unique Solution of Truncated $l_p$ Minimization, Tech. rep., Hong Kong Polytechnic University, 2010.Google Scholar [15] D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. Google Scholar [16] 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 [17] M. Elad and A. M. Bruckstein, A generalized uncertainly priciple and sparse representation in pairs of bases, IEEE Trans. Inf. Theory, 48 (2002), 2558-2567. doi: 10.1109/TIT.2002.801410. Google Scholar [18] G. Fung and O. Mangasarian, Equivalence of minimal $l_0-$ and $l_p-$norm solutions of linear equalities, inequalities and linear programs for sufficiently small $p$, J. Optim. Theory Appl., 151 (2011), 1-10. doi: 10.1007/s10957-011-9871-x. Google Scholar [19] S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $l_q$ minimization for $0 < q < 1$, Applied and Computational Harmonic Analysis, 26 (2009), 395-407. doi: 10.1016/j.acha.2008.09.001. Google Scholar [20] M. A. T. Figueiredo, R. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Select. Top. Signal Process., 1 (2007), 585-597. Google Scholar [21] G. Gasso, A. Rakotomamonjy and S. Canu, Recovering sparse signals with a certain family of nonconvex penalties and DC programming, IEEE Trans. Signal Process., 57 (2009), 4686-4698. doi: 10.1109/TSP.2009.2026004. Google Scholar [22] M. Hyder and K. Mahata, An approximate $l_0$ norm minimization algorithm for compressed sensing, in IEEE International Conference on Acoustics, Speech and Signal Precessing(ICASSP), (2009), 3365-3368.Google Scholar [23] E. T. Hale, W. Yin and Y. Zhang, A fixed-point continuation method for $l_1$-regularized minimization with applications to compressed sensing, CAAM Technical Report TR07-07, Rice University, Houston, TX, 2007.Google Scholar [24] D. Krishnan and R. Fergus, Fast Image Deconvolution Using Hyper-Laplacian Priors, Neural Information Processing Systems., Cambridge, MA: MIT Press, 2009.Google Scholar [25] K. Koh, S.-J. Kim and S. Boyd, The code package l1_ls. Available from: http://www.standord.edu/boyd/l1_ls.Google Scholar [26] Q. Lyu, Z. Lin, Y. She and C. Zhang, A comparison of typical $l_p$ minimization algorithms, Neurocomputing, 119 (2013), 413-424. Google Scholar [27] D. C. Liu and J. Nocedal, On the limited memory method for large scale optimization, Mathematical Programming B, 45 (1989), 503-528. doi: 10.1007/BF01589116. Google Scholar [28] M. Lai and J. Wang, An unconstrained $l_q$ minimization with $0 < q < 1$ for sparse solution of under-determined linear systems, SIAM J. Optim., 21 (2011), 82-101. doi: 10.1137/090775397. Google Scholar [29] B. K. Natraajan, Sparse approximation to linear systems, SIAM J. Comput., 24 (1995), 227-234. doi: 10.1137/S0097539792240406. Google Scholar [30] P. Ochs, A. Dosovitskiy, T. Brox and T. Pock, An iterated $l_1$ algorithm for non-smooth non-convex optimization in computer vision, in Computer Vision and Pattern Recognition (CVPR), IEEE Conference, (2013), 1759-1766.Google Scholar [31] J. K. Pant, W. S. Lu and A. Antoniou, New improved algorithms for compressive sensing based on $l_p$ norm, IEEE Trans. on Circuits and Systems-Ⅱ: Express Briefs, 61 (2014), 198-202. Google Scholar [32] J. Peng, S. Yue and H. Li, NP/CMP equivalence: A phenomenon hidden among sparsity models $l_ {0}$ minimization and $l_ {p}$ minimization for information processing, IEEE Trans. Inf. Theory, 61 (2015), 4028-4033. doi: 10.1109/TIT.2015.2429611. Google Scholar [33] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. Google Scholar [34] Y. She, Thresholding-based iterative selection procedures for model selection and shrinkage, Electron. J. Stat., 3 (2009), 384-415. doi: 10.1214/08-EJS348. Google Scholar [35] Y. She, An iterative algorithm for fitting nonconvex penalized generalized linear models with grouped predictors, Comput. Statist. Data Anal., 9 (2012), 2976-2990. doi: 10.1016/j.csda.2011.11.013. Google Scholar [36] R. Saab, R. Chartrand and O. Yilmaz, Stable sparse approximations via nonconvex optimization, in IEEE International Confereence on Acoustics, Speech and Signal Processing, (2008), 3885-3888.Google Scholar [37] J. Wright, A. Yang, A. Ganesh, S. Sastry and Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Recogn. Anal. Mach. Intell., 31 (2009), 210-227. Google Scholar [38] Y. J. Wang, G. L. Zhou, L. Caccetta and W. Q. Liu, An alternating direction algorithm for $l_1$ problems in compressive sensing, IEEE Trans. Signal Process., 59 (2011), 1895-1901. Google Scholar [39] Y. Wang and Q. Ma, A fast subspace method for image deblurring, Appl. Math. Comput., 215 (2009), 2359-2377. doi: 10.1016/j.amc.2009.08.033. Google Scholar [40] Y. Wang, G. Zhou, X. Zhang, W. Liu and L. Caccetta, The non-convex sparse problem with nonnegative constraint for signal reconstruction, J. Optim. Theory App., 170 (2016), 1009-1025. doi: 10.1007/s10957-016-0869-2. Google Scholar [41] A. Y. Yang, Z. Zhou, A. G. Balasubramanian, S. S. Sastry and Y. Ma, Fast-minimization algorithms for robust face recognition, IEEE Trans. Image Processing, 22 (2013), 3234-3246. Google Scholar [42] F. Zou, H. Feng, H. Ling, C. Liu, L. Yan, P. Li and D. Li, Nonnegative sparse coding induced hashing for image copy detection, Neurocomputing, 105 (2013), 81-95. Google Scholar [43] J. Zeng, S. Lin, Y. Wang and Z. Xu, $L_{1/2}$ regularization: Convergence of iterative half thresholding algorithm, IEEE Trans. Signal Process., 62 (2014), 2317-2329. doi: 10.1109/TSP.2014.2309076. Google Scholar [44] W. Zuo, D. Meng, L. Zhang, X. Feng and D. Zhang, A generalized iterated shrinkage algorithm for non-convex sparse coding, in IEEE International Conference on Computer Vision (ICCV), 2013.Google Scholar

show all references

##### References:
 [1] R. H. Byrd, P. Lu and J. Nocedal, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Stat. Comput., 16 (1995), 1190-1208. doi: 10.1137/0916069. 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. Learning, 3 (2010), 1-122. Google Scholar [3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441. Google Scholar [4] A. Cohen, W. Dahmen and R. DeVore, Compressed sensing and best $k$-term approximation, J. Amer. Math. Soc., 22 (2009), 211-231. doi: 10.1090/S0894-0347-08-00610-3. Google Scholar [5] R. Chartrand, Nonconvex compressed sensing and error correction, IEEE International Conference on Acoustics, Speech and Signal Processing, (2007), 889-892. Google Scholar [6] A. Charkrabarti and F. Hirakawa, Efective separation of sparse and non-sparse image features for denoising, in Proc. Int. Conf. Acoust., Speech, Signal Process. (ICASSP), (2008), 857-860.Google Scholar [7] X. Chen, K. Ng. Michael and C. Zhang, Non-Lipschitz-Regularization and box constrained model for image restoration, IEEE Trans. Image Processing, 21 (2012), 4709-4721. doi: 10.1109/TIP.2012.2214051. Google Scholar [8] E. J. Candès and J. Romberg, The code package $l_1$-magic. Available from: http://statweb.stanford.edu/candes/l1magic/.Google Scholar [9] 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. Google Scholar [10] R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008), 035020, 14 pp. doi: 10.1088/0266-5611/24/3/035020. Google Scholar [11] E. J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979. Google Scholar [12] X. Chen and S. Xiang, Sparse solutions of linear complementarity problems, Math. Program., 159 (2016), 539-556. doi: 10.1007/s10107-015-0950-x. Google Scholar [13] R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing, IEEE International Conference on Acoustics, Speech and Signal Processing, (2008), 3869-3872. Google Scholar [14] X. Chen and W. Zhou, Convergence of Reweighted $l_1$ Minimization Algorithms and Unique Solution of Truncated $l_p$ Minimization, Tech. rep., Hong Kong Polytechnic University, 2010.Google Scholar [15] D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. Google Scholar [16] 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 [17] M. Elad and A. M. Bruckstein, A generalized uncertainly priciple and sparse representation in pairs of bases, IEEE Trans. Inf. Theory, 48 (2002), 2558-2567. doi: 10.1109/TIT.2002.801410. Google Scholar [18] G. Fung and O. Mangasarian, Equivalence of minimal $l_0-$ and $l_p-$norm solutions of linear equalities, inequalities and linear programs for sufficiently small $p$, J. Optim. Theory Appl., 151 (2011), 1-10. doi: 10.1007/s10957-011-9871-x. Google Scholar [19] S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $l_q$ minimization for $0 < q < 1$, Applied and Computational Harmonic Analysis, 26 (2009), 395-407. doi: 10.1016/j.acha.2008.09.001. Google Scholar [20] M. A. T. Figueiredo, R. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Select. Top. Signal Process., 1 (2007), 585-597. Google Scholar [21] G. Gasso, A. Rakotomamonjy and S. Canu, Recovering sparse signals with a certain family of nonconvex penalties and DC programming, IEEE Trans. Signal Process., 57 (2009), 4686-4698. doi: 10.1109/TSP.2009.2026004. Google Scholar [22] M. Hyder and K. Mahata, An approximate $l_0$ norm minimization algorithm for compressed sensing, in IEEE International Conference on Acoustics, Speech and Signal Precessing(ICASSP), (2009), 3365-3368.Google Scholar [23] E. T. Hale, W. Yin and Y. Zhang, A fixed-point continuation method for $l_1$-regularized minimization with applications to compressed sensing, CAAM Technical Report TR07-07, Rice University, Houston, TX, 2007.Google Scholar [24] D. Krishnan and R. Fergus, Fast Image Deconvolution Using Hyper-Laplacian Priors, Neural Information Processing Systems., Cambridge, MA: MIT Press, 2009.Google Scholar [25] K. Koh, S.-J. Kim and S. Boyd, The code package l1_ls. Available from: http://www.standord.edu/boyd/l1_ls.Google Scholar [26] Q. Lyu, Z. Lin, Y. She and C. Zhang, A comparison of typical $l_p$ minimization algorithms, Neurocomputing, 119 (2013), 413-424. Google Scholar [27] D. C. Liu and J. Nocedal, On the limited memory method for large scale optimization, Mathematical Programming B, 45 (1989), 503-528. doi: 10.1007/BF01589116. Google Scholar [28] M. Lai and J. Wang, An unconstrained $l_q$ minimization with $0 < q < 1$ for sparse solution of under-determined linear systems, SIAM J. Optim., 21 (2011), 82-101. doi: 10.1137/090775397. Google Scholar [29] B. K. Natraajan, Sparse approximation to linear systems, SIAM J. Comput., 24 (1995), 227-234. doi: 10.1137/S0097539792240406. Google Scholar [30] P. Ochs, A. Dosovitskiy, T. Brox and T. Pock, An iterated $l_1$ algorithm for non-smooth non-convex optimization in computer vision, in Computer Vision and Pattern Recognition (CVPR), IEEE Conference, (2013), 1759-1766.Google Scholar [31] J. K. Pant, W. S. Lu and A. Antoniou, New improved algorithms for compressive sensing based on $l_p$ norm, IEEE Trans. on Circuits and Systems-Ⅱ: Express Briefs, 61 (2014), 198-202. Google Scholar [32] J. Peng, S. Yue and H. Li, NP/CMP equivalence: A phenomenon hidden among sparsity models $l_ {0}$ minimization and $l_ {p}$ minimization for information processing, IEEE Trans. Inf. Theory, 61 (2015), 4028-4033. doi: 10.1109/TIT.2015.2429611. Google Scholar [33] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. Google Scholar [34] Y. She, Thresholding-based iterative selection procedures for model selection and shrinkage, Electron. J. Stat., 3 (2009), 384-415. doi: 10.1214/08-EJS348. Google Scholar [35] Y. She, An iterative algorithm for fitting nonconvex penalized generalized linear models with grouped predictors, Comput. Statist. Data Anal., 9 (2012), 2976-2990. doi: 10.1016/j.csda.2011.11.013. Google Scholar [36] R. Saab, R. Chartrand and O. Yilmaz, Stable sparse approximations via nonconvex optimization, in IEEE International Confereence on Acoustics, Speech and Signal Processing, (2008), 3885-3888.Google Scholar [37] J. Wright, A. Yang, A. Ganesh, S. Sastry and Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Recogn. Anal. Mach. Intell., 31 (2009), 210-227. Google Scholar [38] Y. J. Wang, G. L. Zhou, L. Caccetta and W. Q. Liu, An alternating direction algorithm for $l_1$ problems in compressive sensing, IEEE Trans. Signal Process., 59 (2011), 1895-1901. Google Scholar [39] Y. Wang and Q. Ma, A fast subspace method for image deblurring, Appl. Math. Comput., 215 (2009), 2359-2377. doi: 10.1016/j.amc.2009.08.033. Google Scholar [40] Y. Wang, G. Zhou, X. Zhang, W. Liu and L. Caccetta, The non-convex sparse problem with nonnegative constraint for signal reconstruction, J. Optim. Theory App., 170 (2016), 1009-1025. doi: 10.1007/s10957-016-0869-2. Google Scholar [41] A. Y. Yang, Z. Zhou, A. G. Balasubramanian, S. S. Sastry and Y. Ma, Fast-minimization algorithms for robust face recognition, IEEE Trans. Image Processing, 22 (2013), 3234-3246. Google Scholar [42] F. Zou, H. Feng, H. Ling, C. Liu, L. Yan, P. Li and D. Li, Nonnegative sparse coding induced hashing for image copy detection, Neurocomputing, 105 (2013), 81-95. Google Scholar [43] J. Zeng, S. Lin, Y. Wang and Z. Xu, $L_{1/2}$ regularization: Convergence of iterative half thresholding algorithm, IEEE Trans. Signal Process., 62 (2014), 2317-2329. doi: 10.1109/TSP.2014.2309076. Google Scholar [44] W. Zuo, D. Meng, L. Zhang, X. Feng and D. Zhang, A generalized iterated shrinkage algorithm for non-convex sparse coding, in IEEE International Conference on Computer Vision (ICCV), 2013.Google Scholar
The original signal and the reconstructed signals with $p = 0.8,0.6,0.4,0.2$, respectively
The original and recovered images with different values of $p$, where SNR denotes the signal-to-noise ratio
Comparison of the performance for ITM and SMM with large-scale problems. The first line: $p = 0.9$; the second line: $p = 0.7$; the final line: $p = 0.5$. The first column: sparsity; the second column: relative error; the final column: cpu time
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