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Self-organizing mappings on the Grassmannian with applications to data analysis in high dimensions

  • WSOM 2017
  • Published:
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Abstract

We propose a method for extending Kohonen’s self-organizing mapping to the geometric framework of the Grassmannian. The resulting algorithm serves as a prototype of the extension of the SOM to the setting of abstract manifolds. The ingredients required for this are a means to measure distance between two points, and a method to move one point in the direction of another. In practice, the data are not required to have a representation in Euclidean space. We discuss in detail how a point on a Grassmannian is moved in the direction of another along a geodesic path. We demonstrate the implementation of the algorithm on several illustrative data sets, hyperspectral images and gene expression data sets.

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Notes

  1. Of course in this example the ordering of the points on the Grassmannian is available to us. However, in general we can determine a one-dimensional parameterization of a set of points on a Grassmannian that approximately passes through nearest neighbors.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant 61771178. We want to thank Shannon Stiverson for discovering the Reactome interferon signal pathway as discriminatory at 48–72 h. This paper is based on research partially supported by the National Science Foundation under Grants No. DMS-1513633, DMS-1322508, and CISE-1712788, as well as DARPA awards N66001-17-2-4020 and D17AP00004.

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Authors and Affiliations

  1. Department of Mathematics, Colorado State University, Fort Collins, CO, USA

    Xiaofeng Ma, Michael Kirby & Chris Peterson

  2. Department of Mathematics and Statistics, Colorado State University, Fort Collins, CO, USA

    Louis Scharf

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  1. Xiaofeng Ma
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  2. Michael Kirby
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  3. Chris Peterson
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  4. Louis Scharf
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Correspondence to Michael Kirby.

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Ma, X., Kirby, M., Peterson, C. et al. Self-organizing mappings on the Grassmannian with applications to data analysis in high dimensions. Neural Comput & Applic 32, 18243–18254 (2020). https://doi.org/10.1007/s00521-019-04444-x

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