# American Institute of Mathematical Sciences

January  2012, 8(1): 163-178. doi: 10.3934/jimo.2012.8.163

## A fast $\ell_1$-solver and its applications to robust face recognition

 1 Sun Yat-sen Univeristy, NO.135 Xiguangxi Road, Guangzhou 510275, China, China, China 2 Curtin University, GPO Box U1987, Perth, WA 6845, Australia, Australia 3 Qufu Normal University, Rizhao 276800, Shandong, China

Received  February 2011 Revised  July 2011 Published  November 2011

In this paper we apply a recently proposed Lagrange Dual Method (LDM) to design a new Sparse Representation-based Classification (LDM-SRC) algorithm for robust face recognition problem. The proposed approach improves the efficiency of the SRC algorithm significantly. The proposed algorithm has the following advantages: (1) it employs the LDM $\ell_1$-solver to find solution of the $\ell_1$-norm minimization problem, which is much faster than other state-of-the-art $\ell_1$-solvers, e.g. $\ell_1$-magic and $\ell_1-\ell_2$. (2) The LDM $\ell_1$-solver utilizes a new Lagrange-dual reformulation of the original $\ell_1$-norm minimization problem, not only reducing the problem size when the dimension of training image data is much less than the number of training samples, but also making the dual problem become smooth and convex. Therefore it converts the non-smooth $\ell_1$-norm minimization problem into a sequence of smooth optimization problems. (3) The LDM-SRC algorithm can maintain good recognition accuracy whilst reducing the computational time dramatically. Experimental results are presented on some benchmark face databases.
Citation: Huining Qiu, Xiaoming Chen, Wanquan Liu, Guanglu Zhou, Yiju Wang, Jianhuang Lai. A fast $\ell_1$-solver and its applications to robust face recognition. Journal of Industrial and Management Optimization, 2012, 8 (1) : 163-178. doi: 10.3934/jimo.2012.8.163
##### References:
 [1] M. S. Bartlett, J. R. Movellan and T. J. Sejnowski, Face recognition by independent component analysis, IEEE Transactions on Neural Networks, 6 (2002), 1450-1464. doi: 10.1109/TNN.2002.804287. [2] M. S. Bazaraa and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," John Wiley & Sons, New York-Chichester-Brisbane, 1979. [3] P. N. Belhumeur, J. Hespanha and D. J. Kriegman, Eigenfaces vs. Fisherfaces: Recognition using class specific linear projection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 7 (1997), 711-720. doi: 10.1109/34.598228. [4] E. van den Berg and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2008/09), 890-912. [5] S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004. [6] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM Journal on Scientific Computing, 16 (1995), 1190-1208. doi: 10.1137/0916069. [7] E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. [8] E. J. Candès and M. Wakin, An introduction to compressive sampling, IEEE Signal Processing Magazine, 2 (2008), 21-30. [9] E. J. Candès and J. Romberg, The code package $\mathcall_1$ -magic: http://www.acm.caltech.edu/l1magic. [10] R. Chellappa, P. Sinha and P. J. Phillips, Face recognition by computers and humans, Computer, 2 (2010), 46-55. doi: 10.1109/MC.2010.37. [11] I. Dagher and R. Nachar, Face recognition using IPCA-ICA algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (2006), 996-1000. doi: 10.1109/TPAMI.2006.118. [12] D. L. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. [13] M. A. T. Figueiredo, R. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, Selected Topics in IEEE Journal of Signal Processing, 4 (2007), 586-597. doi: 10.1109/JSTSP.2007.910281. [14] J.-J. Fuchs, On sparse representations in arbitrary redundant bases, IEEE Transactions on Information Theory, 50 (2004), 1341-1344. doi: 10.1109/TIT.2004.828141. [15] J.-J. Fuchs, Recovery of exact sparse representations in the presence of noise, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'04), 2 (2004), II-533-II-536. [16] J.-J. Fuchs, Fast implementation of a $mathcall_1$-$\mathcall_1$ regularized sparse representations algorithm, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'09), (2009), 3329-3332. [17] X. He, S. Yan, Y. Hu, P. Niyogi and H. Zhang, Face recognition using laplacianfaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 3 (2005), 328-340. [18] K. Koh, S.-J. Kim and S. Boyd, The code package $\mathcall_1-\mathcall_s$: http://www.stanford.edu/~boyd/l1_ls. [19] D. D. Lee and H. S. Seung, Learning the parts of objects by non-negative matrix factorization, Nature, 6755 (1999), 788-791. [20] S. Z. Li, X. Hou, H. Zhang and Q. Cheng, Learning spatially localized, parts-based representation, Proceedings of 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'01), 1 (2001), I-207-I-212. [21] P. J. Phillips, P. J. Flynn, T. Scruggs, K. W. Bowyer, J. Chang, K. Hoffman, J. Marques, J. Min and W. Worek, Overview of the face recognition grand challenge, Proceedings of 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), 1 (2005), 947-954. [22] P. J. Phillips, P. J. Flynn, T. Scruggs, K. W. Bowyer and W. Worek, Preliminary face recognition grand challenge results, The 7th International Conference on Automatic Face and Gesture Recognition (AFGR'06), (2006), 15-24. [23] J. Lu, K. N. Plataniotis and A. N. Venetsanopoulos, Face recognition using LDA-based algorithms, IEEE Transactions on Neural Networks, 1 (2003), 195-200. [24] O. L. Mangasarian and R. R. Meyer, Nonlinear perturbation of linear programs, SIAM Journal on Control and Optimization, 17 (1979), 745-752. doi: 10.1137/0317052. [25] A. M. Martinez and A. C. Kak, PCA versus LDA, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2 (2001), 228-233. doi: 10.1109/34.908974. [26] A. J. O'Toole, P. J. Phillips and A. Narvekar, FRVT 2002 evaluation report, NISTIR TR-6965, 2003. [27] P. J. Phillips, W. T. Scruggs, A. J. O'Toole, P. J. Flynn, K. W. Bowyer, C. L. Schott and M. Sharpe, FRVT 2006 and ICE 2006 large-scale experimental results, IEEE Transactions on Pattern Analysis and Machine Intelligence, 5 (2010), 831-846. doi: 10.1109/TPAMI.2009.59. [28] M. Turk and A. Pentland, Eigenfaces for recognition, Journal of Cognitive Neuroscience, 1 (1991), 71-86. doi: 10.1162/jocn.1991.3.1.71. [29] H.-Y. Wang and X.-J. Wu, Weighted PCA space and its application in face recognition, Proceedings of 2005 International Conference on Machine Learning and Cybernetics, 7 (2005), 4522-4527. [30] X.-G. Wang and X.-O. Tang, A unified framework for subspace face recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9 (2004), 1222-1228. doi: 10.1109/TPAMI.2004.57. [31] Y. Wang, G. Zhou, L. Caccetta and W. Liu, An alternative Lagrange-dual based algorithm for sparse signal reconstruction, IEEE Transactions on Signal Processing, 4 (2011), 1895-1901. doi: 10.1109/TSP.2010.2103066. [32] J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry and Yi Ma, Robust face recognition via sparse representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2 (2009), 210-227. doi: 10.1109/TPAMI.2008.79. [33] S. J. Wright, R. D. Nowak and M. Figueiredo, Sparse reconstruction by separable approximation, IEEE Transactions on Signal Processing, 57 (2009), 2479-2493. doi: 10.1109/TSP.2009.2016892. [34] J. Yang and Y. Zhang, Alternating direction algorithms for $\mathcall_1$ problems in compressive sensing, SIAM Journal on Scientific Computing, 33 (2011), 250-278. doi: 10.1137/090777761. [35] S. Yan, D. Xu, B. Zhang, H.-J. Zhang, Q. Yang and S. Lin, Graph embedding and extensions: A general framework for dimensionality reduction, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1 (2007), 40-51. doi: 10.1109/TPAMI.2007.250598. [36] W. Zhao, R. Chellappa, P. J. Phillips and A. Rosenfeld, Face recognition: A literature survey, ACM Computing Surveys, 4 (2003), 399-458. doi: 10.1145/954339.954342.

show all references

##### References:
 [1] M. S. Bartlett, J. R. Movellan and T. J. Sejnowski, Face recognition by independent component analysis, IEEE Transactions on Neural Networks, 6 (2002), 1450-1464. doi: 10.1109/TNN.2002.804287. [2] M. S. Bazaraa and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," John Wiley & Sons, New York-Chichester-Brisbane, 1979. [3] P. N. Belhumeur, J. Hespanha and D. J. Kriegman, Eigenfaces vs. Fisherfaces: Recognition using class specific linear projection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 7 (1997), 711-720. doi: 10.1109/34.598228. [4] E. van den Berg and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2008/09), 890-912. [5] S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004. [6] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM Journal on Scientific Computing, 16 (1995), 1190-1208. doi: 10.1137/0916069. [7] E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. [8] E. J. Candès and M. Wakin, An introduction to compressive sampling, IEEE Signal Processing Magazine, 2 (2008), 21-30. [9] E. J. Candès and J. Romberg, The code package $\mathcall_1$ -magic: http://www.acm.caltech.edu/l1magic. [10] R. Chellappa, P. Sinha and P. J. Phillips, Face recognition by computers and humans, Computer, 2 (2010), 46-55. doi: 10.1109/MC.2010.37. [11] I. Dagher and R. Nachar, Face recognition using IPCA-ICA algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (2006), 996-1000. doi: 10.1109/TPAMI.2006.118. [12] D. L. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. [13] M. A. T. Figueiredo, R. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, Selected Topics in IEEE Journal of Signal Processing, 4 (2007), 586-597. doi: 10.1109/JSTSP.2007.910281. [14] J.-J. Fuchs, On sparse representations in arbitrary redundant bases, IEEE Transactions on Information Theory, 50 (2004), 1341-1344. doi: 10.1109/TIT.2004.828141. [15] J.-J. Fuchs, Recovery of exact sparse representations in the presence of noise, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'04), 2 (2004), II-533-II-536. [16] J.-J. Fuchs, Fast implementation of a $mathcall_1$-$\mathcall_1$ regularized sparse representations algorithm, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'09), (2009), 3329-3332. [17] X. He, S. Yan, Y. Hu, P. Niyogi and H. Zhang, Face recognition using laplacianfaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 3 (2005), 328-340. [18] K. Koh, S.-J. Kim and S. Boyd, The code package $\mathcall_1-\mathcall_s$: http://www.stanford.edu/~boyd/l1_ls. [19] D. D. Lee and H. S. Seung, Learning the parts of objects by non-negative matrix factorization, Nature, 6755 (1999), 788-791. [20] S. Z. Li, X. Hou, H. Zhang and Q. Cheng, Learning spatially localized, parts-based representation, Proceedings of 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'01), 1 (2001), I-207-I-212. [21] P. J. Phillips, P. J. Flynn, T. Scruggs, K. W. Bowyer, J. Chang, K. Hoffman, J. Marques, J. Min and W. Worek, Overview of the face recognition grand challenge, Proceedings of 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), 1 (2005), 947-954. [22] P. J. Phillips, P. J. Flynn, T. Scruggs, K. W. Bowyer and W. Worek, Preliminary face recognition grand challenge results, The 7th International Conference on Automatic Face and Gesture Recognition (AFGR'06), (2006), 15-24. [23] J. Lu, K. N. Plataniotis and A. N. Venetsanopoulos, Face recognition using LDA-based algorithms, IEEE Transactions on Neural Networks, 1 (2003), 195-200. [24] O. L. Mangasarian and R. R. Meyer, Nonlinear perturbation of linear programs, SIAM Journal on Control and Optimization, 17 (1979), 745-752. doi: 10.1137/0317052. [25] A. M. Martinez and A. C. Kak, PCA versus LDA, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2 (2001), 228-233. doi: 10.1109/34.908974. [26] A. J. O'Toole, P. J. Phillips and A. Narvekar, FRVT 2002 evaluation report, NISTIR TR-6965, 2003. [27] P. J. Phillips, W. T. Scruggs, A. J. O'Toole, P. J. Flynn, K. W. Bowyer, C. L. Schott and M. Sharpe, FRVT 2006 and ICE 2006 large-scale experimental results, IEEE Transactions on Pattern Analysis and Machine Intelligence, 5 (2010), 831-846. doi: 10.1109/TPAMI.2009.59. [28] M. Turk and A. Pentland, Eigenfaces for recognition, Journal of Cognitive Neuroscience, 1 (1991), 71-86. doi: 10.1162/jocn.1991.3.1.71. [29] H.-Y. Wang and X.-J. Wu, Weighted PCA space and its application in face recognition, Proceedings of 2005 International Conference on Machine Learning and Cybernetics, 7 (2005), 4522-4527. [30] X.-G. Wang and X.-O. Tang, A unified framework for subspace face recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9 (2004), 1222-1228. doi: 10.1109/TPAMI.2004.57. [31] Y. Wang, G. Zhou, L. Caccetta and W. Liu, An alternative Lagrange-dual based algorithm for sparse signal reconstruction, IEEE Transactions on Signal Processing, 4 (2011), 1895-1901. doi: 10.1109/TSP.2010.2103066. [32] J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry and Yi Ma, Robust face recognition via sparse representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2 (2009), 210-227. doi: 10.1109/TPAMI.2008.79. [33] S. J. Wright, R. D. Nowak and M. Figueiredo, Sparse reconstruction by separable approximation, IEEE Transactions on Signal Processing, 57 (2009), 2479-2493. doi: 10.1109/TSP.2009.2016892. [34] J. Yang and Y. Zhang, Alternating direction algorithms for $\mathcall_1$ problems in compressive sensing, SIAM Journal on Scientific Computing, 33 (2011), 250-278. doi: 10.1137/090777761. [35] S. Yan, D. Xu, B. Zhang, H.-J. Zhang, Q. Yang and S. Lin, Graph embedding and extensions: A general framework for dimensionality reduction, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1 (2007), 40-51. doi: 10.1109/TPAMI.2007.250598. [36] W. Zhao, R. Chellappa, P. J. Phillips and A. Rosenfeld, Face recognition: A literature survey, ACM Computing Surveys, 4 (2003), 399-458. doi: 10.1145/954339.954342.
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