Topological Grammars for Data Approximation
<|reference_start|>Topological Grammars for Data Approximation: 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.<|reference_end|>
@article{gorban2006topological,
    title={Topological Grammars for Data Approximation},
    author={A.N. Gorban, N.R. Sumner, A.Y. Zinovyev},
    journal={Applied Mathematics Letters 20 (2007) 382--386},
    year={2006},
    doi={10.1016/j.aml.2006.04.022},
    archivePrefix={arXiv},
    eprint={cs/0603090},
    primaryClass={cs.NE cs.LG}
}