On multidimensional scaling and the embedding of self-organising maps

Neural Networks

Volumn 21, Issue 2-3, 2008, Pages 160-169

  Yin, Hujun ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Data visualisation; Dimensionality reduction; Multidimensional scaling; Principal curve and surface; Self organising maps

Indexed keywords

DATA VISUALIZATION; NONLINEAR ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS; PRINCIPAL COMPONENT ANALYSIS; TOPOLOGY;

DIMENSIONALITY REDUCTION; MULTIDIMENSIONAL SCALING (MDS); NONMETRIC SCALING; VISUALISATION INDUCED SOM (VISOM);

SELF ORGANIZING MAPS;

ARTICLE; ARTIFICIAL NEURAL NETWORK; LEARNING ALGORITHM; METRIC SYSTEM; PRIORITY JOURNAL; QUALITATIVE ANALYSIS; SCALE UP;

ALGORITHMS; BRAIN MAPPING; HUMANS; MULTIVARIATE ANALYSIS; NEURAL NETWORKS (COMPUTER); NONLINEAR DYNAMICS; SIGNAL PROCESSING, COMPUTER-ASSISTED;

EID: 40649084778     PISSN: 08936080     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.neunet.2007.12.027     Document Type: Article

References (48)

1. Bauer H.-U., and Pawelzik K.R. Quantifying the neighborhood preservation of self-organizing feature maps. IEEE Transactions on Neural Networks 3 (1992) 570-579
2. Borg I., and Groenen P.J.F. Modern multidimensional scaling: Theory and applications. 2nd ed. (2005), Springer
3. Bronstein A.M., Bronstein M.M., and Kimmel R. Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching. PNAS 103 (2006) 1168-1172
4. Cox T.F., and Cox M.A.A. Multidimensional scaling (1994), Chapman & Hall
5. de Bolt E., Cottrell M., and Verleysen M. Statistical tools to assess the reliability of self-organizing maps. Neural Networks 15 (2002) 967-978
6. Delicado P. Another look at principal curves and surfaces. Journal of Multivariate Analysis 77 (2001) 84-116
7. Durbin R., and Mitchison G. A dimension reduction framework for understanding cortical maps. Nature 343 (1990) 644-647
8. Erwin E., Obermayer K., and Schulten K. Self-organizing maps: Ordering, convergence properties and energy functions. Biological Cybernetics 67 (1992) 47-55
9. Erwin E., Obermayer K., and Schulten K. Self-organizing maps: Stationary states, metastability and convergence rate. Biological Cybernetics 67 (1992) 35-45
10. Estévez P.A., and Figueroa C.J. Online data visualization using the neural gas network. Neural Networks 19 (2006) 923-934
11. Flexer, A. (1997). Limitations of self-organizing maps for vector quantization and multidimensional scaling. In Proc. NIPS'97 (pp. 445-451)
12. Goodhill G.J., and Sejnowski T. A unifying objective function for topographic mappings. Neural Computation 9 (1997) 1291-1303
13. Groenen P.J.F., and Heiser W.J. The tunneling method for global optimization in multidimensional scaling. Psychometrika 61 (1996) 529-550
14. Hastie T., and Stuetzle W. Principal curves. Journal of the American Statistical Association 84 (1989) 502-516
15. Heskes T. Energy functions for self-organizing maps. In: Oja E., and Kaski S. (Eds). Kohonen maps (1999) 303-315
16. Karhunen J., and Joutsensalo J. Generalization of principal component analysis, optimization problems, and neural networks. Neural Networks 8 (1995) 549-562
17. Kohonen T. Self-organized formation of topologically correct feature map. Biological Cybernetics 43 (1982) 56-69
18. Kohonen T. Self-organization and associative memory (1984), Springer-Verlag
19. Kohonen, T. (1986). Representation of sensory information in self-organizing feature maps, and relation of these maps to distributed memory networks. In Proc. SPIE: Vol. 634 (pp. 248-259)
20. Kohonen T. Self-organizing maps: Optimization approaches. In: Kohonen T., Mäkisara K., Simula O., and Kanguas J. (Eds). Artificial neural networks Vol. 2 (1991), North-Holland, Amsterdam 981-990
21. Kohonen T. Self-organizing maps. 2nd ed. (1997), Springer
22. Kramer M.A. Nonlinear principal component analysis using autoassociative neural networks. AICHE Journal 37 (1991) 233-243
23. Lampinen J., and Oja E. Clustering properties of hierarchical self-organizing maps. Journal of Mathematical Imaging and Vision 2 (1992) 261-272
24. Lin S., and Si J. Weight-value convergence of the SOM algorithm for discrete input. Neural Computation 10 (1998) 807-814
25. Lo Z.P., and Bavarian B. On the rate of convergence in topology preserving neural networks. Biological Cybernetics 65 (1991) 55-63
26. Lowe D., and Tipping M.E. Neuroscale: Novel topographic feature extraction using RBF networks. Neural Computing and Applications 4 (1996) 83-95
27. Luttrell S.P. Derivation of a class of training algorithms. IEEE Transactions on Neural Networks 1 (1990) 229-232
28. Luttrell S.P. A Bayesian analysis of self-organizing maps. Neural Computation 6 (1994) 767-794
29. Malthouse E.C. Limitations of nonlinear PCA as performed with generic neural networks. IEEE Transactions on Neural Networks 9 (1998) 165-173
30. Mao J., and Jain A.K. Artificial neural networks for feature extraction and multivariate data projection. IEEE Transactions on Neural Networks 6 (1995) 296-317
31. Mitchison G. A type of duality between self-organizing maps and minimal wiring. Neural Computation 7 (1995) 25-35
32. Ripley B.D. Pattern recognition and neural networks (1996), Cambridge University Press
33. Ritter H., Martinetz T., and Schulten K. Neural computation and self-organizing maps: An introduction (1992), Addison-Wesley Publishing Company
34. Ritter H., and Schulten K. Convergence properties of Kohonen's topology conserving maps: Fluctuations, stability, and dimension selection. Biological Cybernetics 60 (1988) 59-71
35. Roweis S.T., and Saul L.K. Nonlinear dimensionality reduction by locally linear embedding. Science 290 (2000) 2323-2326
36. Sammon J.W. A nonlinear mapping for data structure analysis. IEEE Transactions on Computer 18 (1969) 401-409
37. Schölkopf B., Smola A., and Müller K.R. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 10 (1998) 1299-1319
38. Tenenbaum J.B., de Silva V., and Langford J.C. A global geometric framework for nonlinear dimensionality reduction. Science 290 (2000) 2319-2323
39. Villmann T., Der R., Herrmann M., and Martinetz T.M. Topology preservation in self-organizing feature maps: Exact definition and measurement. IEEE Transactions on Neural Networks 8 (1997) 256-266
40. von der Malsburg C., and Willshaw D. Self-organization of orientation sensitive cells in the striate cortex. Kybernetik 4 (1973) 85-100
41. Willshaw D.J., and von der Malsburg C. How patterned neural connections can be set up by self-organization. Proceedings of the Royal Society of London, Series B 194 (1976) 431-445
42. Wu S., and Chow T.W.S. PRSOM: A new visualization method by hybridizing multidimensional scaling and self-organizing map. IEEE Transactions on Neural Networks 16 (2005) 1362-1380
43. Yin, H. (1996). Self-organizing maps: Statistical analysis, treatment and applications. Ph.D. thesis. University of York
44. Yin H. ViSOM-A novel method for multivariate data projection and structure visualization. IEEE Transactions on Neural Networks 13 (2002) 237-243
45. Yin H. Data visualisation and manifold mapping using the ViSOM. Neural Networks 15 (2002) 1005-1016
46. Yin, H. (2003). Resolution enhancement for the ViSOM. In Proc. workshop on self-organizing maps (pp. 208-212)
47. Yin, H. (2007). Connection between self-organizing maps and metric multidimensional scaling. In Proc. IJCNN 2007 (pp. 1025-1030)
48. Yin H., and Allinson N.M. On the distribution and convergence of the feature space in self-organizing maps. Neural Computation 7 (1995) 1178-1187