Principal graphs and manifolds

Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods, and Techniques

Volumn , Issue , 2009, Pages 28-59

  Gorban, Alexander N ^a,b   Zinovyev, Andrei Y ^c  

^a UNIVERSITY OF LEICESTER   (United Kingdom)

^b ENERGY SYSTEMS INSTITUTE   (Russian Federation)

^c INSTITUT CURIE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84866161460     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.4018/978-1-60566-766-9.ch002     Document Type: Chapter

References (76)

1. Aizerman, M., Braverman, E., & Rozonoer, L. (1964). Theoretical foundations of the potential function method in pattern recognition learning. Automation and Remote Control, 25, 821-837.
2. Aluja-Banet, T., & Nonell-Torrent, R. (1991). Local principal component analysis. Qϋestioό, 3, 267-278.
3. Arthur, D., & Vassilvitskii, S. (2007). K-means++ the advantages of careful seeding. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, LA, USA (pp. 1027-1035).
4. Bau, D., III, & Trefethen, L. N. (1997). Numerical linear algebra. Philadelphia, PA: SIAM Press.
5. Bauer, F. L. (1957). Das verfahren der treppeniteration und verwandte verfahren zur losung algebraischer eigenwert probleme. Math. Phys., 8, 214-235.
6. Belkin, M., & Niyogi, P. (2006). Laplacian Eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6), 1373-1396. doi:10.1162/089976603321780317
7. Bishop, C. M., Svensén, M., & Williams, C. K. I. (1998). GTM: The generative topographic mapping. Neural Computation, 10(1), 215-234. doi:10.1162/089976698300017953
8. Braverman, E. M. (1970). Methods of extremal grouping of parameters and problem of apportionment of essential factors. Automation and Remote Control, (1), 108-116.
9. Cochran, R. N., & Horne, F. H. (1977). Statistically weighted principal component analysis of rapid scanning wavelength kinetics experiments. Analytical Chemistry, 49, 846-853. doi:10.1021/ac50014a045
10. Delicado, P. (2001). Another look at principal curves and surfaces. Journal of Multivariate Analysis, 77, 84-116. doi:10.1006/jmva.2000.1917
11. Dempster, A., Laird, N., & Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B. Methodological, 39(1), 1-38.
12. Diday, E. (1979). Optimisation en classification automatique, Tome 1,2 (in French). Rocquencourt, France: INRIA.
13. Durbin, R., & Willshaw, D. (1987). An analogue approach to the travelling salesman problem using an elastic net method. Nature, 326, 689-691. doi:10.1038/326689a0
14. Efron, B. (1967). The two sample problem with censored data. In Proc. of the Fifth Berkeley Simp. Math. Statist. Probab. 4 (pp. 831-853). Berkeley, CA: Univ.California Press.
15. Einbeck, J., Evers, L., & Bailer-Jones, C. (2008). Representing complex data using localized principal components with application to astronomical data. In A. Gorban, B. Kégl, D. Wunch, & A. Zinovyev (Eds.), Principal manifolds for data visualization and dimension reduction (pp. 178-201). Berlin, Germany; Springer.
16. Einbeck, J., Tutz, G., & Evers, L. (2005). Local principal curves. Statistics and Computing, 15, 301-313. doi:10.1007/s11222-005-4073-8
17. Erwin, E., Obermayer, K., & Schulten, K. (1992). Self-organizing maps: Ordering, convergence properties and energy functions. Biological Cybernetics, 67, 47-55. doi:10.1007/BF00201801
18. Fisher, R. A. (1936). The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7, 179-188.
19. Flury, B. (1990). Principal points. Biometrika, 77, 33-41. doi:10.1093/biomet/77.1.33
20. Fréchet, M. (1948). Les élements aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré, 10, 215-310.
21. Fukunaga, K., & Olsen, D. (1971). An algorithm for finding intrinsic dimensionality of data. IEEE Transactions on Computers, 20(2), 176-183. doi:10.1109/T-C.1971.223208
22. Gabriel, K. R., & Zamir, S. (1979). Lower rank approximation of matrices by least squares with any choices of weights. Technometrics, 21, 298-489. doi:10.2307/1268288
23. Gorban, A., Kégl, B., Wunsch, D., & Zinovyev, A. (Eds.). (2008). Principal manifolds for data visualization and dimension reduction. Berlin, Germany: Springer.
24. Gorban, A., Sumner, N., & Zinovyev, A. (2007). Topological grammars for data approximation. Applied Mathematics Letters, 20(4), 382-386. doi:10.1016/j.aml.2006.04.022
25. Gorban, A., Sumner, N. R., & Zinovyev, A. (2008). Beyond the concept of manifolds: Principal trees, metro maps, and elastic cubic complexes. In A. Gorban, B. Kégl, D. Wunsch, & A. Zinovyev (Eds.), Principal manifolds for data visualization and dimension reduction (pp. 219-237). Berlin, Germany: Springer.
26. Gorban, A., & Zinovyev, A. (2005). Elastic principal graphs and manifolds and their practical applications. Computing, 75, 359-379. doi:10.1007/s00607-005-0122-6
27. Gorban, A., & Zinovyev, A. (2008a). Elastic maps and nets for approximating principal manifolds and their application to microarray data visualization. In A. Gorban, B. Kégl, D. Wunsch, & A. Zinovyev (Eds.), Principal manifolds for data visualization and dimension reduction (pp. 96-130). Berlin, Germany: Springer.
28. Gorban, A., & Zinovyev, A. (2008b). PCA and K-means decipher genome. In A. Gorban, B. Kégl, D. Wunsch, & A. Zinovyev (Eds.), Principal manifolds for data visualization and dimension reduction (pp. 309-323). Berlin, Germnay: Springer.
29. Gorban, A. N., Pitenko, A. A., Zinov'ev, A. Y., & Wunsch, D. C. (2001). Vizualization of any data using elastic map method. Smart Engineering System Design, 11, 363-368.
30. Gorban, A. N., & Rossiev, A. A. (1999). Neural network iterative method of principal curves for data with gaps. Journal of Computer and Systems Sciences International, 38(5), 825-830.
31. Gorban, A. N., Zinovyev, A. Y., & Wunsch, D. C. (2003). Application of the method of elastic maps in analysis of genetic texts. In Proceedings of the International Joint Conference on Neural Networks, IJCNN, Portland, Oregon.
32. Hartigan, J. A. (1975). Clustering algorithms. New York: Wiley.
33. Hastie, T. (1984). Principal curves and surfaces. Unpublished doctoral dissertation, Stanford University, CA.
34. Hotelling, H. (1933). Analisys of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417-441. doi:10.1037/h0071325
35. Ivakhno, S., & Armstrong, J. D. (2007). Non-linear dimensionality reduction of signalling networks. BMC Systems Biology, 1(27).
36. Jolliffe, I. T. (2002). Principal component analysis, series: Springer series in statistics, 2nd ed., XXIX. New York: Springer.
37. Kambhatla, N., & Leen, T. K. (1997). Dimension reduction by local PCA. Neural Computation, 9, 1493-1516. doi:10.1162/neco.1997.9.7.1493
38. Karhunen, K. (1946). Zur spektraltheorie stochastischer prozesse. Suomalainen Tiedeakatemia Toimituksia. Sar. A.4: Biologica, 37.
39. Kégl, B. (1999). Principal curves: Learning, design, and applications. Unpublished doctoral dissertation, Concordia University, Canada.
40. Kégl, B., & Krzyzak, A. (2002). Piecewise linear skeletonization using principal curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(1), 59-74. doi:10.1109/34.982884
41. Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43, 59-69. doi:10.1007/BF00337288
42. Kohonen, T. (1997). Self-organizing maps. Berlin, Germany: Springer.
43. Koren, Y., & Carmel, L. (2004). Robust linear dimensionality reduction. IEEE Transactions on Visualization and Computer Graphics, 10(4), 459-470. doi:10.1109/TVCG.2004.17
44. Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-27. doi:10.1007/BF02289565
45. Leung, Y. F., & Cavalieri, D. (2003). Fundamentals of cDNA microarray data analysis. Trends in Genetics, 19(11), 649-659. doi:10.1016/j.tig.2003.09.015
46. Little, R. J. A., & Rubin, D. B. (1987). Statistical analysis with missing data. New York: John Wiley.
47. Liu, Z.-Y., Chiu, K.-C., & Xu, L. (2003). Improved system for object detection and star/galaxy classification via local subspace analysis. Neural Networks, 16, 437-451. doi:10.1016/S0893-6080(03)00015-7
48. Lloyd, S. (1957). Least square quantization in PCM's (Bell Telephone Laboratories Paper).
49. Loève, M. M. (1955). Probability theory. Princeton, NJ: VanNostrand.
50. Löwe, M. (1993). Algebraic approach to single-pushout graph transformation. Theoretical Computer Science, 109, 181-224. doi:10.1016/0304-3975(93)90068-5
51. Lumley, J. L. (1967). The structure of inhomogeneous turbulent flows. In A. M. Yaglom & V. I. Tatarski (Eds.), Atmospheric turbulence and radio propagation (pp. 166-178). Moscow, Russia: Nauka.
52. th Berkeley Symp. on Math. Statistics and Probability (pp. 281-297).
53. Mirkin, B. (2005). Clustering for data mining: A data recovery approach. Boca Raton, FL: Chapman and Hall.
54. Mulier, F., & Cherkassky, V. (1995). Self-organization as an iterative kernel smoothing process. Neural Computation, 7, 1165-1177. doi:10.1162/neco.1995.7.6.1165
55. Nadler, B., Lafon, S., Coifman, R., & Kevrekidis, I. G. (2008). Diffusion maps - a probabilistic interpretation for spectral embedding and clustering algorithms. In A. Gorban, B. Kégl, D. Wunsch, & A. Zinovyev (Eds.), Principal manifolds for data visualization and dimension reduction (pp. 238-260). Berlin, Germany: Springer.
56. Nagl, M. (1976). Formal languages of labelled graphs. Computing, 16, 113-137. doi:10.1007/BF02241984
57. th Annual IEEE Symposium on Foundations of Computer Science (pp. 165-176). Washington, DC: IEEE Computer Society.
58. Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 6(2), 559-572.
59. Pelleg, D., & Moore, A. (1999). Accelerating exact k-means algorithms with geometric reasoning. In Proceedings of the Fifth International Conference on Knowledge Discovery in Databases (pp. 277-281). Menlo Park, CA: AAAI Press.
60. Ritter, H., Martinetz, T., & Schulten, K. (1992). Neural Computation and Self-Organizing maps: An introduction. Reading, MA: Addison-Wesley.
61. Roweis, S. (1998). EM algorithms for PCA and SPCA. In Advances in neural information processing systems (pp. 626-632). Cambridge, MA: MIT Press.
62. Roweis, S., & Saul, L. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290, 2323-2326. doi:10.1126/science.290.5500.2323
63. Schölkopf, B., Smola, A. J., & Müller, K.-R. (1997). Kernel principal component analysis. In Proceedings of the ICANN (pp. 583-588).
64. Smola, A. J., Mika, S., Schölkopf, B., & Williamson, R. C. (2001). Regularized principal manifolds. Journal of Machine Learning Research, 1, 179-209. doi:10.1162/15324430152748227
65. Steinhaus, H. (1956). Sur la division des corps materiels en parties. In Bull. Acad. Polon. Sci., C1. III vol IV (pp. 801-804).
66. Strang, G. (1993). The fundamental theorem of linear algebra. The American Mathematical Monthly, 100(9), 848-855. doi:10.2307/2324660
67. Sylvester, J. J. (1889). On the reduction of a bilinear quantic of the nth order to the form of a sum of n products by a double orthogonal substitution. Messenger of Mathematics, 19, 42-46.
68. Tarpey, T., & Flury, B. (1996). Self-consistency: A fundamental concept in statistics. Statistical Science, 11(3), 229-243. doi:10.1214/ss/1032280215
69. Tenenbaum, J. B., de Silva, V., & Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality reduction. Science, 290, 2319-2323. doi:10.1126/science.290.5500.2319
70. Tibshirani, R. (1992). Principal curves revisited. Statistics and Computing, 2, 183-190. doi:10.1007/BF01889678
71. Verbeek, J. J., Vlassis, N., & Kröse, B. (2002). A k-segments algorithm for finding principal curves. Pattern Recognition Letters, 23(8), 1009-1017. doi:10.1016/S0167-8655(02)00032-6
72. Wang, Y., Klijn, J. G., & Zhang, Y. (2005). Gene expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer. Lancet, 365, 67-679.
73. Williams, C. K. I. (2002). On a connection between kernel PCA and metric multidimensional scaling. Machine Learning, 46, 11-19. doi:10.1023/A:1012485807823
74. Xu, R., & Wunsch, D. (2008). Clustering. New York: IEEE Press / John Wiley & Sons.
75. Yin, H. (2008). Learning nonlinear principal manifolds by self-organising maps. In A. Gorban, B. Kégl, D. Wunsch, & A. Zinovyev (Eds.), Principal manifolds for data visualization and dimension reduction (pp. 68-95). Berlin, Germany: Springer.
76. Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society, 67(2), 301-320. doi:10.1111/j.1467-9868.2005.00503.x