Topological grammars for data approximation

A method of topological grammars is proposed for multidimensional data approximation. For data with complex topology we define a 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 onedimensional 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...

• Type: Journal
• Year: 2007
• Where: APPML
• Authors: Alexander N. Gorban, Neil R. Sumner, Andrei Yu. Zinovyev
• Keywords: APPML 2007, Complex, Principal Complex, Quadratic Functionals