Dictionary Construction for Patch-to-Tensor Embedding

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

Abstract

The incorporation of matrix relation, which can encompass multidimensional similarities between local neighborhoods of points in the manifold, can improve kernel based data analysis. However, the utilization of multidimensional similarities results in a larger kernel and hence the computational cost of the corresponding spectral decomposition increases dramatically. In this paper, we propose dictionary construction to approximate the kernel in this case and its respected embedding. The proposed dictionary construction is demonstrated on a relevant example of a super kernel that is based on the utilization of the diffusion maps kernel together with linear-projection operators between tangent spaces of the manifold.

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References

  1. 1.
    Baker, C.: The Numerical Treatment of Integral Equations. Clarendon Press, Oxford (1977)MATHGoogle Scholar
  2. 2.
    Coifman, R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21(1), 5–30 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cox, T., Cox, M.: Multidimensional Scaling. Chapman and Hall, London (1994)MATHGoogle Scholar
  4. 4.
    Engel, Y., Mannor, S., Meir, R.: The kernel recursive least-squares algorithm. IEEE Transactions on Signal Processing 52(8), 2275–2285 (2004)MathSciNetCrossRefGoogle 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.
    Singer, A., Wu, H.: Orientability and diffusion maps. Applied and Computational Harmonic Analysis 31(1), 44–58 (2011)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Singer, A., Wu, H.: Vector diffusion maps and the connection laplacian. Communications on Pure and Applied Mathematics 65(8), 1067–1144 (2012)MATHCrossRefGoogle Scholar
  9. 9.
    Wolf, G., Averbuch, A.: Linear-projection diffusion on smooth Euclidean submanifolds. Applied and Computational Harmonic Analysis (2012), doi:10.1016/j.acha, 03.003Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Moshe Salhov
    • 1
  • Guy Wolf
    • 1
  • Amit Bermanis
    • 2
  • Amir Averbuch
    • 1
  • Pekka Neittaanmäki
    • 3
  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Faculty of Information TechnologyUniversity of JyväskyläJyväskyläFinland

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