Fast Parallel Estimation of High Dimensional Information Theoretical Quantities with Low Dimensional Random Projection Ensembles

  • Zoltán Szabó
  • András Lőrincz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5441)

Abstract

The estimation of relevant information theoretical quantities, such as entropy, mutual information, and various divergences is computationally expensive in high dimensions. However, for this task, one may apply pairwise Euclidean distances of sample points, which suits random projection (RP) based low dimensional embeddings. The Johnson-Lindenstrauss (JL) lemma gives theoretical bound on the dimension of the low dimensional embedding. We adapt the RP technique for the estimation of information theoretical quantities. Intriguingly, we find that embeddings into extremely small dimensions, far below the bounds of the JL lemma, provide satisfactory estimates for the original task. We illustrate this in the Independent Subspace Analysis (ISA) task; we combine RP dimension reduction with a simple ensemble method. We gain considerable speed-up with the potential of real-time parallel estimation of high dimensional information theoretical quantities.

Keywords

Independent subspace analysis random projection pairwise distances information theoretical estimations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Shannon, C.: The mathematical theory of communication. Bell System Technical Journal 27, 623–656 (1948)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Neemuchwala, H., Hero, A., Zabuawala, S., Carson, P.: Image registration methods in high-dimensional space. Int. J. Imaging Syst. and Technol. 16(5), 130–145 (2007)CrossRefGoogle Scholar
  3. 3.
    Arya, S., Mount, D.M., Netanyahu, N.S., Silverman, R., Wu, A.Y.: An optimal algorithm for approximate nearest neighbor searching in fixed dimensions. J. of the ACM 45(6), 891 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Johnson, W., Lindenstrauss, J.: Extensions of Lipschitz maps into a Hilbert space. Contemporary Mathematics 26, 189–206 (1984)CrossRefMATHGoogle Scholar
  5. 5.
    Arriga, R.I., Vempala, S.: An algorithmic theory of learning: Robust concepts and random projections. Machine Learning 63, 161–182 (2006)CrossRefGoogle Scholar
  6. 6.
    Matoušek, J.: On variants of the Johnson-Lindenstrauss lemma. Random Structures and Algorithms 33(2), 142–156 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Vempala, S.S.: The Random Projection Method. DIMACS Series in Discrete Math., vol. 65 (2005), http://dimacs.rutgers.edu/Volumes/Vol65.html
  8. 8.
    Baraniuk, R., Davenport, M., DeVore, R., Wakin, M.: A simple proof of the restricted isometry property for random matrices (to appear, 2008)Google Scholar
  9. 9.
    Cardoso, J.: Multidimensional independent component analysis. In: ICASSP 1998, vol. 4, pp. 1941–1944 (1998)Google Scholar
  10. 10.
    Jutten, C., Hérault, J.: Blind separation of sources: An adaptive algorithm based on neuromimetic architecture. Signal Processing 24, 1–10 (1991)CrossRefMATHGoogle Scholar
  11. 11.
    Kybic, J.: High-dimensional mutual information estimation for image registration. In: ICIP 2004, pp. 1779–1782. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  12. 12.
    Gaito, S., Greppi, A., Grossi, G.: Random projections for dimensionality reduction in ICA. Int. J. of Appl. Math. and Comp. Sci. 3(4), 154–158 (2006)Google Scholar
  13. 13.
    Bingham, E.: Advances in Independent Component Analysis with Applications to Data Mining. PhD thesis, Helsinki University of Technology (2003), http://www.cis.hut.fi/ella/thesis/
  14. 14.
    Theis, F.J.: Uniqueness of complex and multidimensional independent component analysis. Signal Processing 84(5), 951–956 (2004)CrossRefMATHGoogle Scholar
  15. 15.
    Theis, F.J.: Multidimensional independent component analysis using characteristic functions. In: EUSIPCO 2005 (2005)Google Scholar
  16. 16.
    Theis, F.J.: Blind signal separation into groups of dependent signals using joint block diagonalization. In: ISCAS 2005, Kobe, Japan, pp. 5878–5881 (2005)Google Scholar
  17. 17.
    Amari, S., Cichocki, A., Yang, H.H.: A new learning algorithm for blind signal separation. In: NIPS 1996, vol. 8, pp. 757–763 (1996)Google Scholar
  18. 18.
    Szabó, Z., Póczos, B., Lőrincz, A.: Cross-entropy optimization for independent process analysis. In: Rosca, J.P., Erdogmus, D., Príncipe, J.C., Haykin, S. (eds.) ICA 2006. LNCS, vol. 3889, pp. 909–916. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Póczos, B., Lőrincz, A.: Independent subspace analysis using k-nearest neighborhood distances. In: Duch, W., Kacprzyk, J., Oja, E., Zadrożny, S. (eds.) ICANN 2005. LNCS, vol. 3697, pp. 163–168. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Szabó, Z., Póczos, B., Lőrincz, A.: Undercomplete blind subspace deconvolution. J. of Machine Learning Res. 8, 1063–1095 (2007)MATHGoogle Scholar
  21. 21.
    Kozachenko, L.F., Leonenko, N.N.: On statistical estimation of entropy of random vector. Problems Infor. Transmiss. 23(2), 95–101 (1987)MATHGoogle Scholar
  22. 22.
    Hero, A., Ma, B., Michel, O., Gorman, J.: Applications of entropic spanning graphs. Signal Processing 19(5), 85–95 (2002)CrossRefGoogle Scholar
  23. 23.
    Rubinstein, R.Y., Kroese, D.P.: The Cross-Entropy Method. Springer, Heidelberg (2004)CrossRefMATHGoogle Scholar
  24. 24.
    Learned-Miller, E.G., Fisher III, J.W.: ICA using spacings estimates of entropy. J. of Machine Learning Res. 4, 1271–1295 (2003)MathSciNetMATHGoogle Scholar
  25. 25.
    Bach, F.R., Jordan, M.I.: Beyond independent components: Trees and clusters. J. of Machine Learning Res. 4, 1205–1233 (2003)MathSciNetMATHGoogle Scholar
  26. 26.
    Póczos, B., Szabó, Z., Kiszlinger, M., Lőrincz, A.: Independent process analysis without a priori dimensional information. In: Davies, M.E., James, C.J., Abdallah, S.A., Plumbley, M.D. (eds.) ICA 2007. LNCS, vol. 4666, pp. 252–259. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Zoltán Szabó
    • 1
  • András Lőrincz
    • 1
  1. 1.Department of Information SystemsEötvös Loránd UniversityBudapestHungary

Personalised recommendations