Fast and Efficient Algorithms for Nonnegative Tucker Decomposition

  • Anh Huy Phan
  • Andrzej Cichocki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5264)

Abstract

In this paper, we propose new and efficient algorithms for nonnegative Tucker decomposition (NTD): Fast α-NTD algorithm which is much precise and faster than α-NTD [1]; and β-NTD algorithm based on the β divergence. These new algorithms include efficient normalization and initialization steps which help to reduce considerably the running time and increase dramatically the performance. Moreover, the multilevel NTD scheme is also presented, allowing further improvements (almost perfect reconstruction). The performance was also compared to other well-known algorithms (HONMF, HOOI, ALS algorithms) for synthetic and real-world data as well.

Keywords

Nonnegative Tucker decomposition (NTD) Nonnegative matrix factorization (NMF) Alpha divergence Beta divergence Hierarchical decomposition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kim, Y.D., Cichocki, A., Choi, S.: Nonnegative Tucker Decomposition with Alpha Divergence. In: 2008 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP2008, Nevada (2008)Google Scholar
  2. 2.
    Tucker, L.R.: Some Mathematical Notes on Three–mode Factor Analysis. Psychometrika 31, 279–311 (1966)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Lathauwer, L.D., Moor, B.D., Vandewalle, J.: A Multilinear Singular Value Decomposition. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Mørup, M., Hansen, L.K., Arnfred, S.M.: Algorithms for Sparse Nonnegative Tucker Decompositions. Neural Computation (in print, 2008)Google Scholar
  5. 5.
    Carroll, J.D., Chang, J.J.: Analysis of Individual Differences in Multidimensional Scaling via an N-way Generalization of Eckart–Young Decomposition. Psychometrika 35, 283–319 (1970)MATHCrossRefGoogle Scholar
  6. 6.
    Phan, A.H., Cichocki, A.: Multi-way Nonnegative Tensor Factorization Using Fast Hierarchical Alternating Least Squares Algorithm (HALS). In: 2008 International Symposium on Nonlinear Theory and its Applications, Budapest (2008)Google Scholar
  7. 7.
    Cichocki, A., Amari, S., Zdunek, R., Kompass, R., Hori, G., He, Z.: Extended SMART Algorithms for Non-Negative Matrix Factorization. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Żurada, J.M. (eds.) ICAISC 2006. LNCS (LNAI), vol. 4029. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Cichocki, A., Zdunek, R., Choi, S., Plemmons, R., Amari, S.: Non-negative Tensor Factorization Using Alpha and Beta Divergences. In: International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2007), pp. 1393–1396. IEEE Press, Honolulu (2007)Google Scholar
  9. 9.
    Lathauwer, L.D., Moor, B.D., Vandewalle, J.: On the Best Rank-1 and Rank-(R1,R2,...,RN) Approximation of Higher-Order Tensors. SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Andersson, C.A., Bro, R.: The N-way Toolbox for MATLAB. Chemometrics and Intelligent Laboratory Systems 52, 1–4 (2000)CrossRefGoogle Scholar
  11. 11.
    Bader, B.W., Kolda, T.G.: MATLAB Tensor Toolbox Version 2.2 (2007), http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/
  12. 12.
    Wu, Q., Xia, T., Yu, Y.: Hierarchical Tensor Approximation of Multi–Dimensional Images. In: 14th IEEE International Conference on Image Processing, vol. 4, pp. 49–52 (2007)Google Scholar
  13. 13.
    Cichocki, A., Zdunek, R.: NMFLAB – NTFLAB for Signal and Image Processing. Technical Report, Laboratory for Advanced Brain Signal Processing, BSI, RIKEN (2006), http://www.bsp.brain.riken.jp
  14. 14.
    The BTF Database Bonn: CEILING Sample, http://btf.cs.uni-bonn.de/download.html

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Anh Huy Phan
    • 1
  • Andrzej Cichocki
    • 1
  1. 1.RIKEN Brain Science InstituteWako-shiJapan

Personalised recommendations