Abstract
Orthogonal nonnegative matrix factorization (NMF) is an NMF objective function that enforces orthogonality constraint on its factor. There are two challenges in optimizing this objective function: the first is how to design an algorithm that has convergence guarantee, and the second is how to automatically choose the regularization parameter. In our previous work, we have been able to develop a convergent algorithm for this objective function. However, the second challenge remains unsolved. In this paper, we provide an attempt to answer the second challenge. The proposed method is based on the L-curve approach and has a simple form which is preferable since it introduces only a small additional computational cost. This method transforms the algorithm into nonparametric, and is also extendable to other NMF objective functions as long as the functions are differentiable with respect to the corresponding regularization parameters. Numerical results are then provided to evaluate the feasibility of the method in choosing the appropriate regularization parameter values by utilizing it in cancer clustering tasks.
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References
Paatero, P., Tapper, U.: Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values. Environmetrics 5, 111–126(1994)
Anttila, P., et al.: Source identification of bulk wet deposition in Finland by positive matrix factorization. Atmospheric Environment 29(14), 1705–1718 (1995)
Lee, D., Seung, H.: Learning the parts of objects by non-negative matrix factorization. Nature 401(6755), 788–791 (1999)
Lee, D., Seung, H.: Algorithms for non-negative matrix factorization. In Proc. Advances in Neural Processing Information Systems, 556–562 (2000)
Xu, W., et al.: Document clustering based on non-negative matrix factorization. In Proc. ACM SIGIR, 267–273 (2003)
Shahnaz, F., et al.: Document clustering using nonnegative matrix factorization. Information Processing & Management 42(2), 373–386 (2006)
Pauca, V.P., et al.: Nonnegative matrix factorization for spectral data analysis. Linear Algebra and Its Applications 416(1), 29–47 (2006)
Jia, S., Qian, Y.: Constrained nonnegative matrix factorization for hyperspectral unmixing. IEEE Transactions on Geoscience and Remote Sensing 47(1), 161–173 (2009)
Li, S.Z., et al.: Learning spatially localized, parts-based representation. In Proc. IEEE Comp. Soc. Conf. on Computer Vision and Pattern Recognition, 207–212 (2001)
Hoyer, P.O.: Non-negative matrix factorization with sparseness constraints. The Journal of Machine Learning Research 5, 1457{1469 (2004)
Wang, D., Lu, H.: On-line learning parts-based representation via incremental orthogonal projective non-negative matrix factorization. Signal Processing 93(6), 1608–1623 (2013)
Pascual-Montano, A., et al.: Nonsmooth nonnegative matrix factorization. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(3), 403{415 (2006)
Gillis, N., Glineur, F.: A multilevel approach for nonnegative matrix factorization. J. Computational and Applied Mathematics 236(7), 1708–1723 (2012)
Cichocki, A., et al.: Extended SMART algorithms for non-negative matrix factorization. LNCS 4029, 548–562, Springer (2006)
Zhou, G., et al.: Online blind source separation using incremental nonnegative matrix factorization with volume constraint. IEEE Transactions on Neural Networks 22(4), 550–560 (2011)
Bertin, N., et al.: Enforcing harmonicity and smoothness in bayesian non-negative matrix factorization applied to polyphonic music transcription. IEEE Transactions on Audio, Speech, and Language Processing 18(3), 538–549 (2010)
Bertrand, A., Moonen, M.: Blind separation of non-negative source signals using multiplicative updates and subspace projection. Signal Processing 90(10), 2877–2890 (2010)
Virtanen, T., et al.: Bayesian extensions to non-negative matrix factorisation for audio signal modelling. In IEEE Int’l Conf. on Acoustics, Speech and Signal Processing, 1825–1828 (2008)
Brunet, J.P., et al.: Metagenes and molecular pattern discovery using matrix factorization. Proc. Natl Acad. Sci. USA 101(12), 4164–4169 (2003)
Gao, Y., Church, G.: Improving molecular cancer class discovery through sparse non-negative matrix factorization. Bioinformatics 21(21), 3970–3975 (2005)
Kim, H., Park, H.: Sparse non-negative matrix factorizations via alternating non-negativity constrained least squares for microarray data analysis. Bioinformatics 23(12), 1495–1502 (2007)
Devarajan, K.: Nonnegative matrix factorization: an analytical and interpretive tool in computational biology. PLoS Computational Biology 4(7), e1000029 (2008)
Kim, H., Park, H.: Nonnegative matrix factorization based on alternating non-negativity constrained least squares and active set method. SIAM J. Matrix Anal. Appl. 30(2), 713–730 (2008)
Carmona-Saez, et al.: Biclustering of gene expression data by non-smooth non-negative matrix factorization. BMC Bioinformatics 7(78) (2006)
Inamura, K., et al.: Two subclasses of lung squamous cell carcinoma with different gene expression profiles and prognosis identified by hierarchical clustering and non-negative matrix factorization. Oncogene (24), 7105–7113 (2005)
Fogel, P., et al.: Inferential, robust non-negative matrix factorization analysis of microarray data. Bioinformatics 23(1), 44–49 (2007)
Zheng, C.H., et al.: Tumor clustering using nonnegative matrix factorization with gene selection. IEEE Transactions on Information Technology in Biomedicine 13(4), 599–607 (2009)
Wang, G., et al.: LS-NMF: A modified non-negative matrix factorization algorithm utilizing uncertainty estimates. BMC Bioinformatics 7(175) (2006)
Wang, J.J.Y., et al.: Non-negative matrix factorization by maximizing correntropy for cancer clustering. BMC Bioinformatics 14(107) (2013)
Yuvaraj, N., Vivekanandan, P.: An efficient SVM based tumor classification with symmetry non-negative matrix factorization using gene expression data. In Int’l Conf. on Information Communication and Embedded Systems, 761–768 (2013)
Ding, C., et al.: Orthogonal nonnegative matrix t-factorizations for clustering. In 12th ACM SIGKDD Int’l Conf. on Knowledge Discovery and Data Mining, 126–135 (2006)
Mirzal, A.: A convergent algorithm for orthogonal nonnegative matrix factorization. To appear in J. Computational and Applied Mathematics.
Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Review 34(4), 561–580 (1992)
Lin, C.J.: On the convergence of multiplicative update algorithms for non-negative matrix factorization. IEEE Transactions on Neural Networks 18(6), 1589–1596 (2007)
Lin, C.J.: Projected gradient methods for non-negative matrix factorization. Technical Report ISSTECH-95-013, Department of CS, National Taiwan University (2005)
Rand, W.M.: Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association 66(336), 846–850 (1971)
Hubert, L., Arabie, P.: Comparing partitions. Journal of Classification 2(1), 193–218 (1985)
Vinh, N.X., et al.: Information theoretic measures for clustering comparison: Is a correction for chance necessary? In 26th Annual Int’l Conf. on Machine Learning, pp. 1073–1080 (2009)
Souto, M.C.P., et al.: Clustering cancer gene expression data: a comparative study. BMC Bioinformatics 9(497) (2008)
Acknowledgments
The author would like to thank the reviewers for useful comments. This research was supported by Ministry of Higher Education of Malaysia and Universiti Teknologi Malaysia under Exploratory Research Grant Scheme R.J130000.7828.4L095.
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Mirzal, A. (2014). Nonparametric Orthogonal NMF and its Application in Cancer Clustering. In: Herawan, T., Deris, M., Abawajy, J. (eds) Proceedings of the First International Conference on Advanced Data and Information Engineering (DaEng-2013). Lecture Notes in Electrical Engineering, vol 285. Springer, Singapore. https://doi.org/10.1007/978-981-4585-18-7_21
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DOI: https://doi.org/10.1007/978-981-4585-18-7_21
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