Abstract
Sparse nonnegative matrix factorizations can be considered as dimension reduction methods that can control the degree of sparseness of basis matrix or coefficient matrix under non-negativity constraints. In this paper, by exploring the sparsity of the basis matrix and the coefficient matrix under certain domains, we propose an alternative iteration approach with l 1-norm minimization for face recognition. Moreover, a modified version of linearized Bregman iteration is developed to efficiently solve the proposed minimization problem. Experimental results show that new algorithm is promising in terms of detection accuracy, computational efficiency.
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References
Cai J.-F., Osher S., Shen Z. (2009a) Linearized Bregman iterations for compressed sensing. Mathematics of Computation 78: 1515–1536
Cai J.-F., Osher S., Shen Z. (2009b) Linearized Bregman iterations for frame-based image deblurring. SIAM Journal on Imaging Sciences 2(1): 226–252
Candès E., Romberg J., Tao T. (2006) Stable signal recovery from incomplete and inaccurate measurements. Communications Pure and Applied Mathematics 59(8): 1207–1223
Cichocki A., Zdunek R., Amari S. I. (2007) Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization. Springer LNCS 4666: 169–176
Cichocki A., Zdunek R., Amari S. (2008) Nonnegative matrix and tensor factorization. IEEE Signal Processing Magazine 25(1): 142–145
Comon P. (1994) Indenpendent component analysis-a new concept?. Signal processing 36: 287–314
Donoho D. (2006) For most large underdetermined systems of linear equations the minimal l 1-norm solution is also the sparsest Solution. Communications Pure and Applied Mathematics 59(6): 797–829
Feng, T., Li, S. Z., & Shum, H.-Y., et al. (2002). Local nonnegative matrix factorization as a visual representation. In Proceedings of second international conference development and learning.
Goldstein, T., & Osher, S. (2008). The split bregman algorithm for L1 regularized problems. UCLA CAM Reports (08-29).
Guillamet D., Vitria J., Schiele B. (2003) Introducing a weighted non-negative matrix factorization for image classification. Pattern Recognition Letters 24: 2447–2454
Hoyer, P. O. (2002). Non-negative sparse coding. In Neural networks for signal processing XII (Proceedings of IEEE workshop on neural networks for signal processing) (pp. 557–565). Martigny, Switzerland.
Hoyer P. O. (2004) Nonnegative matrix factorization with sparseness constraints. Journal of Machine Learning Research 5: 1457–1469
Kim H., Park H. (2007) Sparse non-negative matrix factorizations via alternating non-negativity- constrained least squares for microarray data analysis. Bioinformatics 23(12): 1495–1502
Lee D. D., Seung H. S. (1999) Learning the parts of objects by non-negative matrix factorization. Nature 401(6755): 788–791
Lee D. D., Seung H. S. (2001) Algorithms for non-negative matrix factorization. In Advances in Neural Information Processing 13: 556–562
Liu, W., Zheng, N., & Lu, X. (2003). Non-negative matrix factorization for visual coding. In Proceedings of IEEE international conference on acoustics, speech and signal processing.
Neo, H. F., Teoh, B. J., & Ngo, C. L. (2004). Face recognition using wavelet transform and non-negative matrix factorization. Lecture note on AI, vol. 3339 (pp. 192–202), Berlin: Springer.
Osher S., Burger M., Goldfarb D., Xu J., Yin W. (2005) An iterative regularization method for total variation-based image restoration. Multiscale Model and Simulation 4: 460–489
Osher, S., Mao, Y., Dong, B., & Yin, W. (2008). Fast linearized bregman iteration for compressed sensing and sparse denoising. UCLA CAM Reports (08-37).
Pauca, V. P., & Shahnaz, F., et al. (2004). Text mining using non-negative matrix factorizations. In Proceedings of SIAM international conference data mining, April.
Pauca V. P., Piper J., Plemmons R. J. (2006) Nonnegative matrix factorization for spectral data analysis. Linear Algebra and Applications 416(1): 29–47
Pascual-Montano A., Carazo J. M., Kochi K. et al (2006) Nonsmooth nonnegative matrix factorization (nsNMF). IEEE Transactions on PAMI 28(3): 403–415
Penev P., Atick J. (1996) Local feature analysis: A general statistical theory for object representation. Neural Systems 7(3): 477–500
Tibshirani R. (1996) Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B 58(1): 267–288
Turk M., Pentland A. (1991) Eigenfaces for recognition. Journal of Cognitive Neuroscience 3(1): 71–86
Wright J., Yang A. Y., Ganesh et.al A. (2009) Robust face recognition via sparse representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 31(2): 210–227
Yin W., Osher S., Goldfarb D., Darbon J. (2008) Bregman iterative algorithms for l 1-minimization with applications to compressed sensing. SIAM Journal on Imaging Sciences 1: 143–168
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Jiang, L., Yin, H. Bregman iteration algorithm for sparse nonnegative matrix factorizations via alternating l 1-norm minimization. Multidim Syst Sign Process 23, 315–328 (2012). https://doi.org/10.1007/s11045-011-0147-2
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DOI: https://doi.org/10.1007/s11045-011-0147-2