Topological Grammars for Data Approximation
| dc.creator | Gorban, A. N. | |
| dc.creator | Sumner, N. R. | |
| dc.creator | Zinovyev, A. Y. | |
| dc.date | 2006-03-22 | |
| dc.date | 2006-07-28 | |
| dc.date.accessioned | 2026-07-07T07:05:54Z | |
| dc.date.available | 2026-07-07T07:05:54Z | |
| dc.description | 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. | |
| dc.description | Corrected Journal version, Appl. Math. Lett., in press. 7 pgs., 2 figs | |
| dc.identifier | https://arxiv.org/abs/cs/0603090 | |
| dc.identifier | http://arxiv.org/abs/cs/0603090 | |
| dc.identifier | Applied Mathematics Letters 20 (2007) 382--386 | |
| dc.identifier | doi:10.1016/j.aml.2006.04.022 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/109833 | |
| dc.subject | Neural and Evolutionary Computing | |
| dc.subject | Machine Learning | |
| dc.title | Topological Grammars for Data Approximation | |
| dc.type | text |