Topological grammars for data approximation
Topological grammars for data approximation
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DOI：
10.1016/j.aml.2006.04.022
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发表时间：
2007-04-01
影响因子：
3.7
通讯作者：
Zinovyev, A. Y.
中科院分区：
文献类型：
--
作者：
Gorban, A. N.;Sumner, N. R.;Zinovyev, A. Y.
提出了一种多维数据逼近的拓扑文法方法。对于具有复杂拓扑结构的数据，我们定义了一个低维的主三次复合体，并给出了数据集的最佳近似的复杂性。这个复形是线性和非线性主流形的推广，并将它们作为特例。将最优主复形的构造问题转化为一系列二次泛函的极小化问题。这些二次泛函在弹性能方面有物理上透明的解释。对于能量计算，整个复合体被表示为节点和弹簧的系统。拓扑上，主复形是一维连续体（用图表示）的产物，文法描述了这些连续体在最佳复形构造过程中如何变换。这种分解的整个过程到一维变换使用最小化的二次能量泛函，使我们能够构建有效的算法。(C)2006爱思唯尔有限公司版权所有。
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 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 allows us to construct efficient algorithms. (C) 2006 Elsevier Ltd. All rights reserved.