Geometrical complexity of data approximators

Authors:
Evgeny M. Mirkes;Andrei Zinovyev;Alexander N. Gorban
Affiliations:
Department of Mathematics, University of Leicester, UK;Institut Curie, Paris, France,INSERM U900, Paris, France,Mines ParisTech, Fontainebleau, France;Department of Mathematics, University of Leicester, UK
Venue:
IWANN'13 Proceedings of the 12th international conference on Artificial Neural Networks: advances in computational intelligence - Volume Part I
Year:
2013

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Abstract

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.