Geometrical Complexity of Data Approximators
There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types of principal curves and principal trees, and so on. For each type of approximators the measure of the approximator complexity was developed too. These measures are necessary to find the balance between accuracy and complexity and to define the optimal approximations of a given type. We propose a measure of complexity (geometrical complexity) which is applicable to approximators of several types and which allows comparing data approximations of different types.
KeywordsData analysis Approximation algorithms Data structures Data complexity Model selection
Unable to display preview. Download preview PDF.
- 3.Gorban, A.N., Zinovyev, A.: Principal graphs and manifolds. In: Olivas, E.S., Guererro, J.D.M., Sober, M.M., Benedito, J.R.M., Lopes, A. (eds.) Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Information Science Reference, pp. 28–59. IGI Global, Hershey (2009)Google Scholar
- 4.Zinovyev, A., Mirkes, E.: Data complexity measured by principal graphs. Computers and Mathematics with Applications (2013) doi:10.1016/j.camwa.2012.12.009, arXiv:1212.5841 Google Scholar
- 6.Blakeslee, S.: Lost on earth: wealth of data found in space, An Edward Ng’s quote from the article in New York Times (March 1990)Google Scholar
- 7.Burnham, K.P., Anderson, D.R.: Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd edn. Springer (2002)Google Scholar
- 11.Edmonds, B.: What is complexity? – The philosophy of complexity per se with application to some examples in evolution. In: Heylighen, F., Aerts, D. (eds.) The Evolution of Complexity. Kluwer, Dordrecht (1998)Google Scholar
- 15.Alahakoon, D., Halgamuge, S.K., Sirinivasan, B.: A self growing cluster development approach to data mining. In: Proceedings of IEEE International Conference on Systems, Man and Cybernetics, San Diego, USA, pp. 2901–2906 (1998)Google Scholar
- 16.PCA Master applet, Mirkes, E., University of Leicester (2011) http://bioinfo.curie.fr/projects/elmap
- 17.Kohonen, T.: The Self-Organizing Map (SOM)., http://www.cis.hut.fi/projects/somtoolbox/theory/somalgorithm.shtml
- 18.Gorban, A.N., Kégl, B., Wunch, D.C., Zinovyev, A. (eds.): Principal Manifolds for Data Visualisation and Dimension Reduction. LNSE, vol. 58. Springer, Heidelberg (2008)Google Scholar