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Manifold Learning and Ranking

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Discrete Calculus

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Abstract

A prominent theme of this book is the spatial analysis of networks and data independent of an embedding in an ambient space. The topology and metric of the network/complex have been sufficient to define the domain upon which we may perform data analysis. However, an intrinsic metric defined on a network may be interpreted as the metric that would have been obtained if the network had been embedded into an ambient space equipped with its own metric. Consequently, it is possible to calculate an embedding map for which the induced metric approximates the intrinsic metric defined on the network. The calculation of such embeddings by manifold learning techniques is one way in which the structure of the network may be examined and visualized. A different method of examining the structure of a network is to calculate an importance ranking for each node. In contrast to the majority of this book, the ranking algorithms are generally used to examine the structure of directed graphs.

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Notes

  1. 1.

    Note that Chung defined a symmetric conception of the Laplacian operator on a directed graph [79]. See Chap. 2 for more information on this advection process and the corresponding Laplacian matrix used here.

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Authors and Affiliations

  1. Siemens Corporate Research, 755 College Road East, Princeton, NJ, 08540-6632, USA

    Dr. Leo J. Grady

  2. Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, 02129, USA

    Dr. Jonathan R. Polimeni

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  1. Dr. Leo J. Grady
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  2. Dr. Jonathan R. Polimeni
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Correspondence to Leo J. Grady .

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Grady, L.J., Polimeni, J.R. (2010). Manifold Learning and Ranking. In: Discrete Calculus. Springer, London. https://doi.org/10.1007/978-1-84996-290-2_7

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  • DOI: https://doi.org/10.1007/978-1-84996-290-2_7

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84996-289-6

  • Online ISBN: 978-1-84996-290-2

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