Computer Science – Neural and Evolutionary Computing
Scientific paper
2006-03-22
Applied Mathematics Letters 20 (2007) 382--386
Computer Science
Neural and Evolutionary Computing
Corrected Journal version, Appl. Math. Lett., in press. 7 pgs., 2 figs
Scientific paper
10.1016/j.aml.2006.04.022
A method of {\it topological grammars} is proposed for multidimensional data approximation. For data with complex topology we define a {\it principal cubic complex} of low dimension and given complexity that gives the best approximation for the dataset. This complex is a generalization of linear and non-linear principal manifolds and includes them as particular cases. The problem of optimal principal complex construction is transformed into a series of minimization problems for quadratic functionals. These quadratic functionals have a physically transparent interpretation in terms of elastic energy. For the energy computation, the whole complex is represented as a system of nodes and springs. Topologically, the principal complex is a product of one-dimensional continuums (represented by graphs), and the grammars describe how these continuums transform during the process of optimal complex construction. This factorization of the whole process onto one-dimensional transformations using minimization of quadratic energy functionals allow us to construct efficient algorithms.
Gorban Alexander N.
Sumner N. R.
Zinovyev Andrey Yu.
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