Advertisement

Decomposition in Transition II: Adaptive Tensor Factorization

  • Alexander Paprotny
  • Michael Thess
Chapter
  • 1.7k Downloads
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We consider generalizations of the previously described SVD-based factorization methods to a tensor framework and discuss applications to recommendation. In particular, we generalize the previously introduced incremental SVD algorithm to higher dimensions. Furthermore, we briefly address other tensor factorization frameworks like CANDECOMP/PARAFAC as well as hierarchical SVD and Tensor-Train-Decomposition.

Keywords

Singular Vector Tensor Factorization Canonical Decomposition Hierarchical Decomposition Frontal Mode 
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.

References

  1. [DLDMV00]
    De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  2. [DSL08]
    De Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084–1127 (2008)CrossRefMathSciNetGoogle Scholar
  3. [Gra10]
    Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. [HK09]
    Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706–722 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. [KABO10]
    Karatzoglou, A., Amatriain, X., Baltrunas, L., Oliver, N.: Multiverse recommendation: n-dimensional tensor factorization for context-aware collaborative filtering. In: Proceedings of the Fourth ACM Conference on Recommender Systems, RecSys ’10, pp. 79–86. ACM, New York (2010)Google Scholar
  6. [KB09]
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. [Os11]
    Oseledets, I.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. [OST08]
    Oseledets, I., Savostyanov, D., Tyrtyshnikov, E.: Tucker dimensionality reduction of three-dimensional arrays in linear time. SIAM J. Matrix Anal. Appl. 30(3), 939–956 (2008)CrossRefMathSciNetGoogle Scholar
  9. [OT09]
    Oseledets, I., Tyrtyshnikov, E.: Recursive and tensor-train decompositions in higher dimensions. In: Proceedings of The 9th Hellenic European Research on Computer Mathematics & its Applications Conference (2009)Google Scholar
  10. [OT10]
    Oseledets, I., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432, 70–88 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. [RFST10]
    Rendle, S., Freudenthaler, C., Schmidt-Thieme L.: Factorizing personalized Markov chains for next-basket recommendation. In: Proceedings of the 19th International World Wide Web Conference (WWW 2010), ACM (2010)Google Scholar
  12. [SO11]
    Savostyanov, D., Oseledets, I.: Fast adaptive interpolation of multi-dimensional arrays in tensor train format. In: Proceedings of 7th International Workshop on Multidimensional Systems (nDS), IEEE (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Alexander Paprotny
    • 1
  • Michael Thess
    • 2
  1. 1.Research and Developmentprudsys AGBerlinGermany
  2. 2.Research and Developmentprudsys AGChemnitzGermany

Personalised recommendations