Regularized nonnegative matrix factorization: Geometrical interpretation and application to spectral unmixing

Benthem, M.H.V. and Keenan, M.R. (2004). Fast algorithm for the solution of large-scale non-negativity-constrained least squares problems, Journal of Chemometrics18(10): 441– 450.10.1002/cem.889Search in Google Scholar

Berry, M., Browne, M., Langville, A.N., Pauca, P. and Plemmons, R.J. (2007). Algorithms and applications for approximate nonnegative matrix factorization, Computational Statistics and Data Analysis52(1): 155–173.10.1016/j.csda.2006.11.006Search in Google Scholar

Bioucas-Dias, J.M. (2009). A variable splitting augmented Lagrangian approach to linear spectral unmixing, Proceedings of the 1st IEEE GRSS Workshop on Hyperspectral Image and Signal Processing, WHISPERS, Grenoble, France.10.1109/WHISPERS.2009.5289072Search in Google Scholar

Bioucas-Dias, J.M. and Figueiredo, M. (2010). Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing, Proceedings of the 2nd IEEE GRSS Workshop on Hyperspectral Image and Signal Processing, WHISPERS, Raykjavik, Iceland.10.1109/WHISPERS.2010.5594963Search in Google Scholar

Bioucas-Dias, J.M., Plaza, A., Dobigeon, N., Parente, M., Du, Q., Gader, P. and Chanussot, J. (2012). Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing5(2): 354–379.10.1109/JSTARS.2012.2194696Search in Google Scholar

Bro, R. and Jong, S.D. (1997). A fast non-negativityconstrained least squares algorithm, Journal of Chemometrics11(5): 393–401.10.1002/(SICI)1099-128X(199709/10)11:5<393::AID-CEM483>3.0.CO;2-LSearch in Google Scholar

Calvetti, D., Lewis, B. and Reichel, L. (2001). On the choice of subspace for iterative methods for linear discrete ill-posed problems, International Journal of Applied Mathematics and Computer Science11(5): 1069–1092.Search in Google Scholar

Chan, T.-H., Ma, W.-K., Ambikapathi, A.-M. and Chi, C.-Y. (2011). A simplex volume maximization framework for hyperspectral endmember extraction, IEEE Transactions on Geoscience and Remote Sensing49(11): 4177–4193.10.1109/TGRS.2011.2141672Search in Google Scholar

Chen, D. and Plemmons, R.J. (2009). Nonnegativity constraints in numerical analysis, in A. Bultheel and R. Cools (Eds.), The Birth of Numerical Analysis, World Scientific, Singapore, pp. 109–139.10.1142/9789812836267_0008Search in Google Scholar

Chu, M.T. and Lin, M. M. (2008). Low dimensional polytype approximation and its applications to nonnegative matrix factorization, SIAM Journal of Scientific Computing30(3): 1131–1151.10.1137/070680436Search in Google Scholar

Cichocki, A., Zdunek, R., Phan, A.H. and Amari, S.-I. (2009). Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation, Wiley and Sons, Chichester.10.1002/9780470747278Search in Google Scholar

Dabrowski, A. and Cetnarowicz, D. (2008). Iterative SVD algorithm as a BSS solution, Proceedings of the International Conference on Signals and Electronic Systems, IC-SES 2008, Cracow, Poland, pp. 401–404.Search in Google Scholar

Donoho, D. and Stodden, V. (2004). When does non-negative matrix factorization give a correct decomposition into parts?, in S. Thrun, L. Saul and B. Sch¨olkopf (Eds.), Advances in Neural Information Processing Systems (NIPS), Vol. 16, MIT Press, Cambridge, MA, pp. 1141–1148.Search in Google Scholar

Elden, L. (1977). Algorithms for the regularization of illconditioned least squares problems, BIT17(2): 134–145.10.1007/BF01932285Search in Google Scholar

Garda, B. and Galias, Z. (2012). Non-negative least squares and the Tikhonov regularization methods for coil design problems, Proceedings of the International Conference on Signals and Electronic Systems, ICSES’12, Wrocław, Poland.10.1109/ICSES.2012.6382220Search in Google Scholar

Gobinet, C., Perrin, E. and Huez, R. (2004). Application of nonnegative matrix factorization to fluorescence spectroscopy, Proceedings of the European Signal Processing Conference, EUSIPCO 2004, Vienna, Austria, pp. 1095–1098.Search in Google Scholar

Górecki, T. and Łuczak, M. (2013). Linear discriminant analysis with a generalization of the Moore–Penrose pseudoinverse, International Journal of Applied Mathematics and Computer Science23(2): 463–471, DOI: 10.2478/amcs-20130035.Search in Google Scholar

Guo, Z., Wittman, T. and Osher, S. (2009). L1 unmixing and its application to hyperspectral image enhancement, in S.S. Shen and P.E. Lewis (Eds.), Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery XV, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7334, International Society for Optical Engineering, Orlando, FL, p. 73341M+.10.1117/12.818245Search in Google Scholar

Hamza, A. and Brady, D. (2006). Reconstruction of reflectance spectra using robust nonnegative matrix factorization, IEEE Transactions on Signal Processing54(9): 3637– 3642.10.1109/TSP.2006.879282Search in Google Scholar

Hancewicz, T.M. and Wang, J.-H. (2005). Discriminant image resolution: A novel multivariate image analysis method utilizing a spatial classification constraint in addition to bilinear nonnegativity, Chemometrics and Intelligent Laboratory Systems77(1–2): 18–31.10.1016/j.chemolab.2004.07.013Search in Google Scholar

Hansen, P.C. (1998). Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, PA.10.1137/1.9780898719697Search in Google Scholar

Heylen, R., Burazerovic, D. and Scheunders, P. (2011). Fully constrained least squares spectral unmixing by simplex projection, IEEE Transactions on Geoscience and Remote Sensing49(11): 4112–4122.10.1109/TGRS.2011.2155070Search in Google Scholar

Huck, A., Guillaume, M. and Blanc-Talon, J. (2010). Minimum dispersion constrained nonnegative matrix factorization to unmix hyperspectral data, IEEE Transactions on Geoscience and Remote Sensing48(6): 2590–2602.10.1109/TGRS.2009.2038483Search in Google Scholar

Hyvrinen, A., Karhunen, J. and Oja, E. (2001). Independent Component Analysis, John Wiley, New York, NY.10.1002/0471221317Search in Google Scholar

Igual, J. and Llinares, R. (2008). Nonnegative matrix factorization of laboratory astrophysical ice mixtures, IEEE Journal of Selected Topics in Signal Processing2(5): 697–706.10.1109/JSTSP.2008.2005324Search in Google Scholar

Igual, J., Llinares, R. and Salazar, A. (2006). Source separation of astrophysical ice mixtures, Proceedings of the 6th International Conference on Independent Component Analysis and Blind Signal Separation, Charleston, IL, USA, pp. 368–375.Search in Google Scholar

Iordache, M., Dias, J. and Plaza, A. (2011). Sparse unmixing of hyperspectral data, IEEE Transactions on on Geoscience and Remote Sensing49(2): 2014–2039.10.1109/TGRS.2010.2098413Search in Google Scholar

Iordache, M., Dias, J. and Plaza, A. (2012). Total variation spatial regularization for sparse hyperspectral unmixing, IEEE Transactions on Geoscience and Remote Sensing50(11): 4484–4502.10.1109/TGRS.2012.2191590Search in Google Scholar

Jia, S. and Qian, Y. (2009). Constrained nonnegative matrix factorization for hyperspectral unmixing, IEEE Transactions on Geoscience and Remote Sensing47(1): 161–173.10.1109/TGRS.2008.2002882Search in Google Scholar

Kim, D., Sra, S. and Dhillon, I.S. (2007). Fast Newtontype methods for the least squares nonnegative matrix approximation problem, Proceedings of the 6th SIAM International Conference on Data Mining, Minneapolis, MN, USA, pp. 343–354.Search in Google Scholar

Kim, H. and Park, H. (2008). Non-negative matrix factorization based on alternating non-negativity constrained least squares and active set method, SIAM Journal on Matrix Analysis and Applications30(2): 713–730.10.1137/07069239XSearch in Google Scholar

Kim, J. and Park, H. (2011). Fast nonnegative matrix factorization: An active-set-like method and comparisons, SIAM Journal on Scientific Computing33(6): 3261–3281.10.1137/110821172Search in Google Scholar

Krawczyk-Sta´ndo, D. and Rudnicki, M. (2007). Regularization parameter selection in discrete ill-posed problems— The use of the U-curve, International Journal of Applied Mathematics and Computer Science17(2): 157–164, DOI: 10.2478/v10006-007-0014-3.10.2478/v10006-007-0014-3Search in Google Scholar

Kulczycki, P. and Charytanowicz, M. (2010). A complete gradient clustering algorithm formed with kernel estimators, International Journal of Applied Mathematics and Computer Science20(1): 123–134, DOI: 10.2478/v10006-010-0009-3.10.2478/v10006-010-0009-3Search in Google Scholar

Lawson, C.L. and Hanson, R.J. (1974). Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, NJ.Search in Google Scholar

Lee, D.D. and Seung, H.S. (1999). Learning the parts of objects by non-negative matrix factorization, Nature401(6755): 788–791.10.1038/4456510548103Search in Google Scholar

Li, H., Adali, T., Wang, W., Emge, D. and Cichocki, A. (2007). Non-negative matrix factorization with orthogonality constraints and its application to Raman spectroscopy, The Journal of VLSI Signal Processing48(1–2): 83–97.10.1007/s11265-006-0039-0Search in Google Scholar

Li, J. and Bioucas-Dias, J.M. (2008). Minimum volume simplex analysis: A fast algorithm to unmix hyperspectral data, Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, IGARSS 2008, Boston, MA, USA, Vol. 3, pp. 250–253.Search in Google Scholar

Likic, V.A. (2009). Extraction of pure components from overlapped signals in gas chromatography-mass spectrometry (GC-MS), BioData Mining2(6): 1–11.10.1186/1756-0381-2-6277054919818154Search in Google Scholar

Lin, C.-J. (2007). Projected gradient methods for non-negative matrix factorization, Neural Computation19(10): 2756– 2779.10.1162/neco.2007.19.10.275617716011Search in Google Scholar

Llinares, R., Igual, J., Mir´o-Borr´as, J. and Camacho, A. (2010). Analysis of astrophysical ice analogs using regularized alternating least squares, Proceedings of the 20th International Conference on Artificial Neural Networks, ICANN 2010, Thessaloniki, Greece, pp. 199–204.Search in Google Scholar

Makowski, R. (2003). Source pulse estimation of mine shocks by blind deconvolution, Pure and Applied Geophysics160(7): 1191–1205.10.1007/s000240300001Search in Google Scholar

Miao, L. and Qi, H. (2007). Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization, IEEE Transactions on Geoscience and Remote Sensing45(3): 765–777.10.1109/TGRS.2006.888466Search in Google Scholar

Miron, S., Dossot, M., Carteret, C., Margueron, S. and Brie, D. (2011). Joint processing of the parallel and crossed polarized Raman spectra and uniqueness in blind nonnegative source separation, Chemometrics and Intelligent Laboratory Systems105(1): 7–18.10.1016/j.chemolab.2010.10.005Search in Google Scholar

Nascimento, J.M.P. and Bioucas-Dias, J.M. (2005). Vertex component analysis: A fast algorithm to unmix hyperspectral data, IEEE Transactions on Geoscience and Remote Sensing43(4): 898–910.10.1109/TGRS.2005.844293Search in Google Scholar

Pauca, V.P., Pipera, J. and Plemmons, R.J. (2006). Nonnegative matrix factorization for spectral data analysis, Linear Algebra and Its Applications416(1): 29–47.10.1016/j.laa.2005.06.025Search in Google Scholar

Pengo, T., Munoz-Barrutia, A. and de Solorzano, C.O. (2010). Spectral unmixing of multiply stained fluorescence samples, in A. Mendez-Vilas and J. Diaz (Eds.), Microscopy: Science, Technology, Applications and Education, Microscopy Book Series, No. 4, Formatex Research Center, Badajoz, pp. 2079–2087.Search in Google Scholar

Plaza, J., Hendrix, E.M.T., Garc´ıa, I., Martin, G. and Plaza, A. (2012). On endmember identification in hyperspectral images without pure pixels: A comparison of algorithms, Journal of Mathematical Imaging and Vision42(2– 3): 163–175.10.1007/s10851-011-0276-0Search in Google Scholar

Qian, Y., Jia, S., Zhou, J. and Robles-Kelly, A. (2011). Hyperspectral unmixing via l1/2 sparsity-constrained nonnegative matrix factorization, IEEE Transactions on Geoscience and Remote Sensing49(11): 4282–4297.10.1109/TGRS.2011.2144605Search in Google Scholar

Rojas, M. and Steihaug, T. (2002). An interior-point trustregion-based method for large-scale non-negative regularization, Inverse Problems18(5): 1291–1307.10.1088/0266-5611/18/5/305Search in Google Scholar

Sajda, P., Du, S., Brown, T., Parra, L. and Stoyanova, R. (2003). Recovery of constituent spectra in 3D chemical shift imaging using nonnegative matrix factorization, Proceedings of the 4th International Symposium on Independent Component Analysis and Blind Signal Separation, Nara, Japan, pp. 71–76.Search in Google Scholar

Sajda, P., Du, S., Brown, T.R., Stoyanova, R., Shungu, D.C., Mao, X. and Parra, L.C. (2004). Nonnegative matrix factorization for rapid recovery of constituent spectra in magnetic resonance chemical shift imaging of the brain, IEEE Transactions on Medical Imaging23(12): 1453–1465.10.1109/TMI.2004.83462615575404Search in Google Scholar

Siwek, K., Osowski, S. and Szupiluk, R. (2009). Ensemble neural network approach for accurate load forecasting in a power system, International Journal of Applied Mathematics and Computer Science19(2): 303–315, DOI: 10.2478/v10006-009-0026-2.10.2478/v10006-009-0026-2Search in Google Scholar

Tong, L., van der Veen, A.-J., Dewilde, P. and Sung, Y. (2003). Blind decorrelating RAKE receivers for longcode WCDMA, IEEE Transactions on Signal Processing51(6): 1642–1655.10.1109/TSP.2003.811230Search in Google Scholar

Wózniak, M. and Krawczyk, B. (2012). Combined classifier based on feature space partitioning, International Journal of Applied Mathematics and Computer Science22(4): 855– 866, DOI: 10.2478/v10006-012-0063-0.10.2478/v10006-012-0063-0Search in Google Scholar

Zdunek, R. (2011). Regularized active set least squares algorithm for nonnegative matrix factorization in application to Raman spectra separation, in J. Cabestany, I. Rojas and G. Joya (Eds.), Advances in Computational Intelligence, Lecture Notes in Computer Science, Vol. 6692, Springer, Berlin/Heidelberg, pp. 492–499.10.1007/978-3-642-21498-1_62Search in Google Scholar

Zdunek, R. (2012). Hyperspectral image unmixing with nonnegative matrix factorization, Proceedings of the IEEE International Conference on Signals and Electronic Systems, ICSES 2012, Wrocław, Poland.10.1109/ICSES.2012.6382219Search in Google Scholar

Zdunek, R. and Cichocki, A. (2007). Nonnegative matrix factorization with constrained second-order optimization, Signal Processing87(8): 1904–1916.10.1016/j.sigpro.2007.01.024Search in Google Scholar

Zhang, J., Rivard, B. and Rogge, D.M. (2008). The successive projection algorithm (SPA), an algorithm with a spatial constraint for the automatic search of endmembers in hyperspectral data, Sensors8(2): 1321–1342.10.3390/s8021321392751227879768Search in Google Scholar

Zymnis, A., Kim, S.-J., Skaf, J., Parente, M. and Boyd, S. (2007). Hyperspectral image unmixing via alternating projected subgradients, Proceedings of the 41st Asilomar Conference on Signals, Systems and Computers, ACSSC 2007, Pacific Grove, CA, USA, pp. 1164–1168.Search in Google Scholar