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Self-Organizing Mappings on the Flag Manifold

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Advances in Self-Organizing Maps, Learning Vector Quantization, Clustering and Data Visualization (WSOM 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 976))

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Abstract

A flag is a nested sequence of vector spaces. The type of the flag is determined by the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a particular type in a fixed vector space. This paper provides the mathematical framework necessary for implementing self-organizing mappings on flag manifolds. Flags arise implicitly in many data analysis techniques for instance in wavelet, Fourier, and singular value decompositions. The proposed geometric framework in this paper enables the computation of distances between flags, the computation of geodesics between flags, and the ability to move one flag a prescribed distance in the direction of another flag. Using these operations as building blocks, we implement the SOM algorithm on a flag manifold. The basic algorithm is applied to the problem of parameterizing a set of flags of a fixed type.

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References

  1. Daubechies I (1992) Ten lectures on wavelets. CBMS-NSF regional conference series in applied mathematics. SIAM, Philadelphia

    Google Scholar 

  2. Draper B, Kirby M, Marks J, Marrinan T, Peterson C (2014) A flag representation for finite collections of subspaces of mixed dimensions. Linear Algebra Appl 451:15–32

    Article  MathSciNet  Google Scholar 

  3. Alan E, Arias TA, Smith ST (1998) The geometry of algorithms with orthogonality constraints. SIAM J Matrix Anal Appl 20(2):303–353

    Article  MathSciNet  Google Scholar 

  4. Kaski S, Kangas J, Kohonen T (1998) Bibliography of self-organizing map (SOM) papers: 1981–1997. Neural Comput Surv 1(3&4):1–176

    Google Scholar 

  5. Kirby M (2001) Geometric data analysis: an empirical approach to dimensionality reduction and the study of patterns. Wiley, New York

    MATH  Google Scholar 

  6. Kirby, M, Peterson, C (2017) Visualizing data sets on the Grassmannian using self-organizing mappings. In: 2017 12th international workshop on self-organizing maps and learning vector quantization, clustering and data visualization (WSOM), June 2017, pp 1–6

    Google Scholar 

  7. Kohonen T (1982) Self-organized formation of topologically correct feature maps. Biol Cybern 43:59

    Article  MathSciNet  Google Scholar 

  8. Kohonen T (1990) The self-organizing map. Proc IEEE 78(9):1464–1480

    Article  Google Scholar 

  9. Kohonen T (1998) The self-organizing map. Neurocomputing 21(1):1–6

    Article  MathSciNet  Google Scholar 

  10. Kohonen T (2013) Essentials of the self-organizing map. Neural Netw 37:52–65

    Article  Google Scholar 

  11. Ma, X, Kirby, M, Peterson, C, Scharf, L (2018) Self-organizing mappings on the Grassmannian with applications to data analysis in high-dimensions. Neural Comput Appl

    Google Scholar 

  12. Mallat S (1989) Multiresolution approximations and wavelet orthonormal bases of \(l^2\). Trans Am Math Soc 315:69–87

    Google Scholar 

  13. Mallat S (1989) A theory of multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 315:69–87

    MATH  Google Scholar 

  14. Tim, M, Beveridge, JR, Draper, B, Kirby, M, Peterson, C (2015) Flag manifolds for the characterization of geometric structure in large data sets. In: Numerical mathematics and advanced applications-ENUMATH. Springer, pp 457–465

    Google Scholar 

  15. Tim, M, Bruce, D, Beveridge, JR, Kirby, M, Peterson, C (2014) Finding the subspace mean or median to fit your need. In: Computer Vision and Pattern Recognition (CVPR). IEEE, pp 1082–1089

    Google Scholar 

  16. Marrinan, T, Beveridge, JR, Draper, B, Kirby, M, Peterson, C (2016) Flag-based detection of weak gas signatures in long-wave infrared hyperspectral image sequences. In: Proceedings SPIE Defense + Security. International Society for Optics and Photonics, p 98401N

    Google Scholar 

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Author information

Authors and Affiliations

  1. Colorado State University, Fort Collins, CO, 80523, USA

    Xiaofeng Ma, Michael Kirby & Chris Peterson

Authors
  1. Xiaofeng Ma
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  2. Michael Kirby
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  3. Chris Peterson
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Corresponding author

Correspondence to Michael Kirby .

Editor information

Editors and Affiliations

  1. Department of Computer Science, UPC BarcelonaTech, Barcelona, Spain

    Alfredo Vellido

  2. Knowledge Engineering and Machine Learning Group (KEMLG) at Intelligent Data Science and Artificial Intelligence Research Center, UPC BarcelonaTech, Barcelona, Spain

    Karina Gibert

  3. Department of Automatic Control, UPC BarcelonaTech, Barcelona, Spain

    Cecilio Angulo

  4. Departament d'Enginyeria Electrònica, Universitat de València, Burjassot, Valencia, Spain

    José David Martín Guerrero

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Ma, X., Kirby, M., Peterson, C. (2020). Self-Organizing Mappings on the Flag Manifold. In: Vellido, A., Gibert, K., Angulo, C., Martín Guerrero, J. (eds) Advances in Self-Organizing Maps, Learning Vector Quantization, Clustering and Data Visualization. WSOM 2019. Advances in Intelligent Systems and Computing, vol 976. Springer, Cham. https://doi.org/10.1007/978-3-030-19642-4_2

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