Abstract
The solution for the parameters of a nonlinear mapping in a metric multidimensional scaling by transformation, in which a stress criterion is optimised, satisfies a nonlinear eigenvector equation, which may be solved iteratively. This can be cast in a kernel-based framework in which the configuration of training samples in the transformation space may be found iteratively by successive linear projections, without the need for gradient calculations. A new data sample can be projected using knowledge of the kernel and the final configuration of data points.
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J. M. Borwein and A. S. Lewis. Convex Analysis and Nonlinear Optimisation. Theory and Examples. Springer-Verlag, New York, 2000.
N. Cristianini and J. Shawe-Taylor. An introduction to support vector machines. Cambridge University Press, Cambridge, 2000.
W. R. Dillon and M. Goldstein. Multivariate Analysis Methods and Applications. John Wiley and Sons, New York, 1984.
W. J. Heiser. Convergent computation by iterative majorization: theory and applications in multidimensional data analysis. In W. J. Krzanowski, editor, Recent Advances in Descriptive Multivariate Analysis, pages 157–189. Clarendon Press, Oxford, 1994.
W. L. G. Koontz and K. Fukunaga. A nonlinear feature extraction algorithm using distance information. IEEE Transactions on Computers, 21(1):56–63, 1972.
B. Lerner, H. Guterman, M. Aladjem, and I. Dinstein. A comparative study of neural network based feature extraction paradigms. Pattern Recognition Letters, 120:7–14, 1999.
B. Lerner, H. Guterman, M. Aladjem, I. Dinstein, and Y. Romem. On pattern classification with Sammon’s nonlinear mapping-an experimental study. Pattern Recognition, 31(4):371–381, 1998.
D. Lowe and M. Tipping. Feed-forward neural networks and topographic mappings for exploratory data analysis. Neural Computing and Applications, 4:83–95, 1996.
J. Mao and A. K. Jain. Artificial neural networks for feature extraction and multivariate data projection. IEEE Transactions on Neural Networks, 6(2):296–317, 1995.
R. Mathar and R. Meyer. Algorithms in convex analysis to fit lp-distance matrices. Journal of Multivariate Analysis, 51:102–120, 1994.
R. Meyer. Nonlinear eigenvector algorithms for local optimisation in multivariate data analysis. Linear Algebra and its Applications, 264:225–246, 1997.
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes. The Art of Scientific Computing. Cambridge University Press, Cambridge, second edition, 1992.
R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, New Jersey, 1970.
J. W. Sammon. A nonlinear mapping for data structure analysis. IEEE Transactions on Computers, 18(5):401–409, 1969.
B. Schölkopf, A. Smola, and K.-R. Müller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299–1319, 1998.
A. R. Webb. Multidimensional scaling by iterative majorisation using radial basis functions. Pattern Recognition, 28(5):753–759, 1995.
A. R. Webb. Radial basis functions for exploratory data analysis: an iterative majorisation approach for Minkowski distances based on multidimensional scaling. Journal of Classification, 14(2):249–267, 1997.
A. R. Webb. Statistical Pattern Recognition. Arnold, London, 1999.
A. R. Webb. A kernel approach to metric multidimensional scaling. In preparation, 2002.
C. K. I. Williams. On a connection between kernel pca and metric multidimensional scaling. Machine Learning, 46(1/3):11–19, 2001.
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Webb, A. (2002). A Kernel Approach to Metric Multidimensional Scaling. In: Caelli, T., Amin, A., Duin, R.P.W., de Ridder, D., Kamel, M. (eds) Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2002. Lecture Notes in Computer Science, vol 2396. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-70659-3_47
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DOI: https://doi.org/10.1007/3-540-70659-3_47
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