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Principal Manifolds for Data Visualization and Dimension Reduction pp 96–130Cite as

Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization

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

Principal manifolds are defined as lines or surfaces passing through “the middle” of data distribution. Linear principal manifolds (Principal Components Analysis) are routinely used for dimension reduction, noise filtering and data visualization. Recently, methods for constructing non-linear principal manifolds were proposed, including our elastic maps approach which is based on a physical analogy with elastic membranes. We have developed a general geometric framework for constructing “principal objects” of various dimensions and topologies with the simplest quadratic form of the smoothness penalty which allows very effective parallel implementations. Our approach is implemented in three programming languages (C++, Java and Delphi) with two graphical user interfaces (VidaExpert and ViMiDa applications). In this paper we overview the method of elastic maps and present in detail one of its major applications: the visualization of microarray data in bioinformatics. We show that the method of elastic maps outperforms linear PCA in terms of data approximation, representation of between-point distance structure, preservation of local point neighborhood and representing point classes in low-dimensional spaces.

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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

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

    Andrei Y. Zinovyev

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  1. Alexander N. Gorban
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  2. 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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Gorban, A.N., Zinovyev, A.Y. (2008). Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization. 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_4

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