Patch-Based Data Analysis Using Linear-Projection Diffusion

  • Moshe Salhov
  • Guy Wolf
  • Amir Averbuch
  • Pekka Neittaanmäki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7619)

Abstract

To process massive high-dimensional datasets, we utilize the underlying assumption that data on a manifold is approximately linear in sufficiently small patches (or neighborhoods of points) that are sampled with sufficient density from the manifold. Under this assumption, each patch can be represented by a tangent space of the manifold in its area and the tangential point of this tangent space. We use these tangent spaces, and the relations between them, to extend the scalar relations that are used by many kernel methods to matrix relations, which can encompass multidimensional similarities between local neighborhoods of points on the manifold. The properties of the presented construction are explored and its spectral decomposition is utilized to embed the patches of the manifold into a tensor space in which the relations between them are revealed. We present two applications that utilize the patch-to-tensor embedding framework: data classification and data clustering for image segmentation.

Keywords

Dimensionality reduction manifold learning kernel PCA Diffusion Maps patch processing stochastic processing vector processing 

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References

  1. 1.
    Coifman, R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21(1), 5–30 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Cox, T., Cox, M.: Multidimensional Scaling. Chapman and Hall, London (1994)MATHGoogle Scholar
  3. 3.
    Frank, A., Asuncion, A.: UCI machine learning repository (2010), http://archive.ics.uci.edu/ml
  4. 4.
    Jossinet, J.: Variability of impedivity in normal and pathological breast tissue. Medical and Biological Engineering and Computing 34, 346–350 (1996)CrossRefGoogle Scholar
  5. 5.
    Kruskal, J.: Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29, 1–27 (1964)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Salhov, M., Wolf, G., Averbuch, A.: Patch-to-tensor embedding. Applied and Computational Harmonic Analysis 33(2), 182–203 (2012)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    da Silva, J.E., de Sá, J.M., Jossinet, J.: Classification of breast tissue by electrical impedance spectroscopy. Medical and Biological Engineering and Computing 38, 26–30 (2000)CrossRefGoogle Scholar
  8. 8.
    Singer, A., Wu, H.: Orientability and diffusion maps. Applied and Computational Harmonic Analysis 31(1), 44–58 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Singer, A., Wu, H.: Vector diffusion maps and the connection laplacian. Communications on Pure and Applied Mathematics 65(8), 1067–1144 (2012)MATHCrossRefGoogle Scholar
  10. 10.
    Wolf, G., Averbuch, A.: Linear-projection diffusion on smooth Euclidean submanifolds. Applied and Computational Harmonic Analysis (2012), doi:10.1016/j.acha.2012.03.003Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Moshe Salhov
    • 1
  • Guy Wolf
    • 1
  • Amir Averbuch
    • 1
  • Pekka Neittaanmäki
    • 2
  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Faculty of Information TechnologyUniversity of JyväskyläJyväskyläFinland

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