Journal of KIISE:Computing Practices and Letters (한국정보과학회논문지:컴퓨팅의 실제 및 레터)

Korean Institute of Information Scientists and Engineers (한국정보과학회)
권호 표지

Nonnegative Tucker Decomposition

텐서의 비음수 Tucker 분해

  • 김용덕 (포항공과대학교 컴퓨터공학과) ;
  • 최승진 (포항공과대학교 컴퓨터공학과)
  • Published : 2008.05.15
Download PDF
⟨ Previous Next ⟩

Abstract

Nonnegative tensor factorization(NTF) is a recent multiway(multilineal) extension of nonnegative matrix factorization(NMF), where nonnegativity constraints are imposed on the CANDECOMP/PARAFAC model. In this paper we consider the Tucker model with nonnegativity constraints and develop a new tensor factorization method, referred to as nonnegative Tucker decomposition (NTD). We derive multiplicative updating algorithms for various discrepancy measures: least square error function, I-divergence, and $\alpha$-divergence.

최근에 개발된 Nonnegative tensor factorization(NTF)는 비음수 행렬 분해(NMF)의 multiway(multilinear) 확장형이다. NTF는 CANDECOMP/PARAFAC 모델에 비음수 제약을 가한 모델이다. 본 논문에서는 Tucker 모델에 비음수 제약을 가한 nonnegative Tucker decomposition(NTD)라는 새로운 텐서 분해 모델을 제안한다. 제안된 NTD 모델을 least squares, I-divergence, $\alpha$-divergence를 이용한 여러 목적함수에 대하여 fitting하는 multiplicative update rule을 유도하였다.

Keywords

References

  1. A. Hyv¨arinen, J. Karhunen, and E. Oja, "Independent Component Analysis," John Wiley & Sons, Inc., 2001
  2. D. D. Lee and H. S. Seung, "Learning the parts of objects by non-negative matrix factorization," Nature, 401:788.791, 1999 https://doi.org/10.1038/44565
  3. J. Yang, D. Zhang, A. F. Frangi, and J. Y. Yang, "Tw-o dimensional PCA: A new approach to appearance-based face representation and recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, 26(1):131.137, 2004 https://doi.org/10.1109/TPAMI.2004.1261097
  4. D. Zhang, S. Chen, and Z. H. Zhou, "Two-dimensional non-negative matrix factorization for face representation and recognition," In ICCV-2005 Workshop on Analysis and Modeling of Faces and Gestures, 2005
  5. L. R. Tucker, "Some mathematical notes on three- mode factor analysis," Psychometrika, 31:279.311, 1966 https://doi.org/10.1007/BF02289464
  6. L. de Lathauwer, B. de Moor, and J. Vandewalle, "A multilinear singular value decomposition," SIAM J. Matrix Anal. Appl., 21(4):1253.1278, 2000 https://doi.org/10.1137/S0895479896305696
  7. R. A. Harshman, "Foundations of the PARAFAC procedure: Models and conditions for an Exploratory. multi-modal factor analysis," UCLA Working Papers in Phonetics, 16:1.84, 1970
  8. A. Shashua and T. Hazan, "Non-negative tensor facto-rization with applications to statistics and computer vision," In Proceedings of International Conference on Machine Learning, Bonn, Germany, 2005
  9. A. Cichocki, R. Zdunek, S. Choi, R. J. Plemmons, and S. Amari, "Non-negative tensor factorization using alpha and beta divergences," In Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, Honolulu, Hawaii, 2007
  10. A. Pascual-Montano, J. M. Carazo, K. K. D. Lehmann, and R. D. Pascual-Margui, "Nonsmooth nonnegtive matrix factorization (nsNMF)," IEEE Trans. Pattern Analysis and Machine Intelligence, 28(3):403.415, 2006 https://doi.org/10.1109/TPAMI.2006.60