Advertisement

Projective Nonnegative Matrix Factorization with α-Divergence

  • Zhirong Yang
  • Erkki Oja
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5768)

Abstract

A new matrix factorization algorithm which combines two recently proposed nonnegative learning techniques is presented. Our new algorithm, α-PNMF, inherits the advantages of Projective Nonnegative Matrix Factorization (PNMF) for learning a highly orthogonal factor matrix. When the Kullback-Leibler (KL) divergence is generalized to α-divergence, it gives our method more flexibility in approximation. We provide multiplicative update rules for α-PNMF and present their convergence proof. The resulting algorithm is empirically verified to give a good solution by using a variety of real-world datasets. For feature extraction, α-PNMF is able to learn highly sparse and localized part-based representations of facial images. For clustering, the new method is also advantageous over Nonnegative Matrix Factorization with α-divergence and ordinary PNMF in terms of higher purity and smaller entropy.

Keywords

Facial Image Nonnegative Matrix Factorization Classical Principal Component Analysis Multiplicative Update Principal Component Analysis Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401, 788–791 (1999)CrossRefGoogle Scholar
  2. 2.
    Amari, S.: Differential-geometrical methods in statistics. Lecture Notes in Statistics, vol. 28. Springer, New York (1985)zbMATHGoogle Scholar
  3. 3.
    Cichocki, A., Lee, H., Kim, Y.D., Choi, S.: Non-negative matrix factorization with α-divergence. Pattern Recognition Letters 29, 1433–1440 (2008)CrossRefGoogle Scholar
  4. 4.
    Yuan, Z., Oja, E.: Projective nonnegative matrix factorization for image compression and feature extraction. In: Kalviainen, H., Parkkinen, J., Kaarna, A. (eds.) SCIA 2005. LNCS, vol. 3540, pp. 333–342. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Phillips, P.J., Moon, H., Rizvi, S.A., Rauss, P.J.: The FERET evaluation methodology for face recognition algorithms. IEEE Trans. Pattern Analysis and Machine Intelligence 22, 1090–1104 (2000)CrossRefGoogle Scholar
  6. 6.
    Ding, C., Li, T., Peng, W., Park, H.: Orthogonal nonnegative matrix t-factorizations for clustering. In: Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 126–135 (2006)Google Scholar
  7. 7.
    Brunet, J.P., Tamayo, P., Golub, T.R., Mesirov, J.P.: Metagenes and molecular pattern discovery using matrix factorization. Proceedings of the National Academy of Sciences 101(12), 4164–4169 (2004)CrossRefGoogle Scholar
  8. 8.
    Samaria, F., Harter, A.: Parameterisation of a stochastic model for human face identification. In: Proceedings of 2nd IEEE Workshop on Applications of Computer Vision, Sarasota FL, December 1994, pp. 138–142 (1994)Google Scholar
  9. 9.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Zhirong Yang
    • 1
  • Erkki Oja
    • 1
  1. 1.Department of Information and Computer ScienceHelsinki University of TechnologyEspooFinland

Personalised recommendations