Advertisement

Fast Similarity Computation in Factorized Tensors

  • Michael E. Houle
  • Hisashi Kashima
  • Michael Nett
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7404)

Abstract

Low-rank factorizations of higher-order tensors have become an invaluable tool for researchers from many scientific disciplines. Tensor factorizations have been successfully applied for moderately sized multimodal data sets involving a small number of modes. However, a significant hindrance to the full realization of the potential of tensor methods is a lack of scalability on the client side: even when low-rank representations are provided by an external agent possessing the necessary computational resources, client applications are quickly rendered infeasible by the space requirements for explicitly storing a (dense) low-rank representation of the input tensor. We consider the problem of efficiently computing common similarity measures between entities expressed by fibers (vectors) or slices (matrices) within a given factorized tensor. We show that after appropriate preprocessing, inner products can be efficiently computed independently of the dimensions of the input tensor.

Keywords

similarity computation inner products tensor factorization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andersson, C.A., Henrion, R.: A general algorithm for obtaining simple structure of core arrays in n-way PCA with application to fluorometric data. Computational Statistics & Data Analysis 31(3), 255–278 (1999)zbMATHCrossRefGoogle Scholar
  2. 2.
    Appellof, C.J., Davidson, E.R.: Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents. Analytical Chemistry 53(13), 2053–2056 (1981)CrossRefGoogle Scholar
  3. 3.
    Bader, B.W., Kolda, T.G.: Efficient MATLAB computations with sparse and factored tensors. SIAM Journal on Scientific Computing 30(1), 205–231 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bader, B.W., Kolda, T.G.: MATLAB Tensor Toolbox Version 2.4 (March 2011), http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/
  5. 5.
    Bennett, J., Lanning, S., Netflix: The Netflix Prize. In: KDD Cup and Workshop in conjunction with KDD (2007)Google Scholar
  6. 6.
    Cantador, I., Brusilovsky, P., Kuflik, T.: 2nd workshop on information heterogeneity and fusion in recommender systems (HetRec 2011). In: Proceedings of the 5th ACM Conference on Recommender Systems, RecSys 2011, ACM (2011)Google Scholar
  7. 7.
    Carroll, J., Chang, J.-J.: Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition. Psychometrika 35, 283–319 (1970)zbMATHCrossRefGoogle Scholar
  8. 8.
    Chen, B., Petropulu, A.P., Lathauwer, L.D.: Blind identification of convolutive MIMO systems with 3 sources and 2 sensors. EURASIP Journal on Advances in Signal Processing 5, 487–496 (2002)CrossRefGoogle Scholar
  9. 9.
    Chi, Y., Tseng, B.L., Tatemura, J.: Eigen-trend: trend analysis in the blogosphere based on singular value decompositions. In: Proceedings of the 15th ACM International Conference on Information and Knowledge Management, CIKM, pp. 68–77 (2006)Google Scholar
  10. 10.
    Comon, P.: Independent component analysis, a new concept? Signal Processing 36(3), 287–314 (1994)zbMATHCrossRefGoogle Scholar
  11. 11.
    Dunlavy, D.M., Kolda, T.G., Acar, E.: Temporal link prediction using matrix and tensor factorizations. ACM Transactions on Knowledge Discovery from Data 5(2), Article 10, 27 pages (2011)CrossRefGoogle Scholar
  12. 12.
    Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1, 211–218 (1936)zbMATHCrossRefGoogle Scholar
  13. 13.
    GroupLens Research Group. GroupLens Research Data Sets (August 2011), http://www.grouplens.org/node/12
  14. 14.
    Håstad, J.: Tensor rank is NP-complete. Journal of Algorithms 11(4), 644–654 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Harshman, R.A.: Foundations of the PARAFAC procedure: models and conditions for an explanatory multi-modal factor analysis. UCLA Working Papers in Phonetics 16, 1–84 (1970)Google Scholar
  16. 16.
    Hitchcock, F.L.: The expression of a tensor or a polyadic as a sum of products. J. Math. 6, 164–189 (1927)zbMATHGoogle Scholar
  17. 17.
    Kapteyn, A., Neudecker, H., Wansbeek, T.: An approach to n-mode components analysis. Psychometrika 51, 269–275 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Karatzoglou, A., Amatriain, X., Baltrunas, L., Oliver, N.: Multiverse recommendation: n-dimensional tensor factorization for context-aware collaborative filtering. In: Proceedings of the 4th ACM Conference on Recommender Systems, pp. 79–86 (2010)Google Scholar
  19. 19.
    Kolda, T., Bader, B.: The TOPHITS model for higher-order web link analysis. In: Proceedings of the SIAM Data Mining Conference Workshop on Link Analysis, Counterterrorism and Security (2006)Google Scholar
  20. 20.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Review 51(3), 455–500 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kolda, T.G., Bader, B.W., Kenny, J.P.: Higher-order web link analysis using multilinear algebra. In: ICDM 2005: Proceedings of the 5th IEEE International Conference on Data Mining, pp. 242–249 (November 2005)Google Scholar
  22. 22.
    Kolda, T.G., Sun, J.: Scalable tensor decompositions for multi-aspect data mining. In: ICDM 2008: Proceedings of the 8th IEEE International Conference on Data Mining, pp. 363–372 (December 2008)Google Scholar
  23. 23.
    Lin, Z.: Some software packages for partial SVD computation. Computing Research Repository (CoRR) abs/1108.1548 (2011)Google Scholar
  24. 24.
    Möcks, J.: Topographic components model for event-related potentials and some biophysical considerations. IEEE Transactions on Biomedical Engineering 35(6), 482–484 (1988)CrossRefGoogle Scholar
  25. 25.
  26. 26.
    Rendle, S., Schmidt-Thieme, L.: Pairwise interaction tensor factorization for personalized tag recommendation. In: WSDM, pp. 81–90 (2010)Google Scholar
  27. 27.
    Ren, Henrion: N-way principal component analysis theory, algorithms and applications. Chemometrics and Intelligent Laboratory Systems 25(1), 1–23 (1994)CrossRefGoogle Scholar
  28. 28.
    Smilde, A.K., Wang, Y., Kowalski, B.R.: Theory of medium-rank second-order calibration with restricted-Tucker models. Journal of Chemometrics 8(1), 21–36 (1994)CrossRefGoogle Scholar
  29. 29.
    Tucker, L.: Some mathematical notes on three-mode factor analysis. Psychometrika 31, 279–311 (1966)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Vasilescu, M.A.O., Terzopoulos, D.: Multilinear Analysis of Image Ensembles: TensorFaces. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002, Part I. LNCS, vol. 2350, pp. 447–460. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michael E. Houle
    • 1
  • Hisashi Kashima
    • 2
    • 3
  • Michael Nett
    • 1
    • 2
  1. 1.National Institute of InformaticsTokyoJapan
  2. 2.University of TokyoTokyoJapan
  3. 3.Basic Research Programs PRESTOSynthesis of Knowledge for Information Oriented SocietyJapan

Personalised recommendations