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Principal Manifolds for Data Visualization and Dimension Reduction pp 219–237Cite as

Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes

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Part of the book series: Lecture Notes in Computational Science and Enginee ((LNCSE,volume 58))

Multidimensional data distributions can have complex topologies and variable local dimensions. To approximate complex data, we propose a new type of low-dimensional “principal object”: a principal cubic complex. This complex is a generalization of linear and non-linear principal manifolds and includes them as a particular case. To construct such an object, we combine a method of topological grammars with the minimization of an elastic energy defined for its embedment into multidimensional data space. The whole complex is presented as a system of nodes and springs and as a product of one-dimensional continua (represented by graphs), and the grammars describe how these continua transform during the process of optimal complex construction

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Alexander N. Gorban & Neil R. Sumner

  2. Institut Curie Service Bioinformatique, rue d'Ulm 26, 75248, Paris, France

    Andrei Y. Zinovyev

Authors
  1. Alexander N. Gorban
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  2. Neil R. Sumner
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  3. Andrei Y. Zinovyev
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Editor information

Editors and Affiliations

  1. University of Leicester, University Road, LE1 7RH, Leicester, UK

    Alexander N. Gorban

  2. University of Paris-Sud - CNRS, Bâtiment 2000, 91898, Orsay, France

    Balázs Kégl

  3. University of Missouri - Rolla, 1870 Miner Circle, 65409, Rolla, MO, USA

    Donald C. Wunsch

  4. Institut Curie Service Bioinformatique, rue d'Ulm 26, 75248, Paris, France

    Andrei Y. Zinovyev

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© 2008 Springer-Verlag Berlin Heidelberg

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Gorban, A.N., Sumner, N.R., Zinovyev, A.Y. (2008). Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes. In: Gorban, A.N., Kégl, B., Wunsch, D.C., Zinovyev, A.Y. (eds) Principal Manifolds for Data Visualization and Dimension Reduction. Lecture Notes in Computational Science and Enginee, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73750-6_9

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