Local multidimensional scaling. (English) Zbl 1102.68601

Summary: In a visualization task, every nonlinear projection method needs to make a compromise between trustworthiness and continuity. In a trustworthy projection the visualized proximities hold in the original data as well, whereas a continuous projection visualizes all proximities of the original data. We show experimentally that one of the multidimensional scaling methods, curvilinear components analysis, is good at maximizing trustworthiness. We then extend it to focus on local proximities both in the input and output space, and to explicitly make a user-tunable parameterized compromise between trustworthiness and continuity. The new method compares favorably to alternative nonlinear projection methods.


68T05 Learning and adaptive systems in artificial intelligence
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[1] Belkin, M.; Niyogi, P., Laplacian eigenmaps and spectral techniques for embedding and clustering, (Dietterich, T. G.; Becker, S.; Ghahramani, Z., Advances in neural information processing systems, vol. 14 (2002), MIT Press: MIT Press Cambridge, MA), 585-591
[4] Borg, I.; Groenen, P., Modern multidimensional scaling (1997), Springer: Springer New York · Zbl 0862.62052
[5] Demartines, P.; Hérault, J., Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets, IEEE Transactions on Neural Networks, 8, 148-154 (1997)
[6] Gower, J. C., Some distance properties of latent root and vector methods used in multivariate analysis, Biometrika, 53, 325-338 (1966) · Zbl 0192.26003
[7] Hinton, G.; Roweis, S., Stochastic neighbor embedding, (Advances in neural information processing systems, vol. 15 (2002), MIT Press), 833-840
[8] Hotelling, H., Analysis of a complex of statistical variables into principal components, Journal of Educational Psychology, 24, 417-441 (1933), 498-520
[9] Kaski, S.; Nikkilä, J.; Oja, M.; Venna, J.; Törönen, P.; Castrén, E., Trustworthiness and metrics in visualizing similarity of gene expression, BMC Bioinformatics, 4, 48 (2003)
[10] Kohonen, T., Self-organizing maps (2001), Springer: Springer Berlin · Zbl 0957.68097
[11] Kruskal, J. B., Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis, Psychometrica, 29, 1-26 (1964) · Zbl 0123.36803
[12] Lee, J. A.; Lendasse, A.; Donckers, N.; Verleysen, M., A robust nonlinear projection method, (Verleysen, M., ESANN’2000, Eighth European symposium on artificial neural networks (2000), D-Facto Publications: D-Facto Publications Bruges, Belgium), 13-20
[13] Lee, J. A.; Lendasse, A.; Verleysen, M., Nonlinear projection with curvilinear distances: Isomap versus curvilinear distance analysis, Neurocomputing, 57, 49-76 (2004)
[14] Roweis, S. T.; Saul, L. K., Nonlinear dimensionality reduction by locally linear embedding, Science, 290, 2323-2326 (2000)
[15] Sammon, J. W., A nonlinear mapping for data structure analysis, IEEE Transactions on Computers, C-18, 401-409 (1969)
[16] Segal, E.; Koller, N. F.A. D.; Regev, A., A module map showing conditional activity of expression modules in cancer, Nature Genetics, 36, 1090-1098 (2004)
[17] Su, A. I.; Cooke, M. P.; Ching, K. A.; Hakak, Y.; Walker, J. R.; Wiltshire, T., Large-scale analysis of the human and mouse transcriptomes, Proceedings of the National Academy of Sciences, 99, 4465-4470 (2002)
[18] Tenenbaum, J. B.; de Silva, V.; Langford, J. C., A global geometric framework for nonlinear dimensionality reduction, Science, 290, 2319-2323 (2000)
[19] Torgerson, W. S., Multidimensional scaling: I. Theory and method, Psychometrika, 17, 401-419 (1952) · Zbl 0049.37603
[20] Ultsch, A., Self-organization neural networks for visualization and classification, (Opitz, O.; Lausen, B.; Klar, R., Information and classification (1993), Springer-Verlag: Springer-Verlag Berlin), 307-313
[21] Venna, J.; Kaski, S., Neighborhood preservation in nonlinear projection methods: An experimental study, (Dorffner, G.; Bischof, H.; Hornik, K., Proceedings of ICANN 2001, international conference on artificial neural networks (2001), Springer: Springer Berlin), 485-491 · Zbl 1001.68746
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