Skip to main content

Tensor Factorization Using Auxiliary Information

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNAI,volume 6912)


Most of the existing analysis methods for tensors (or multi-way arrays) only assume that tensors to be completed are of low rank. However, for example, when they are applied to tensor completion problems, their prediction accuracy tends to be significantly worse when only limited entries are observed. In this paper, we propose to use relationships among data as auxiliary information in addition to the low-rank assumption to improve the quality of tensor decomposition. We introduce two regularization approaches using graph Laplacians induced from the relationships, and design iterative algorithms for approximate solutions. Numerical experiments on tensor completion using synthetic and benchmark datasets show that the use of auxiliary information improves completion accuracy over the existing methods based only on the low-rank assumption, especially when observations are sparse.


  • Link Prediction
  • Auxiliary Information
  • Tensor Factorization
  • Tensor Decomposition
  • Sylvester Equation

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.


  1. Acar, E., Dunlavy, D.M., Kolda, T.G., Mørup, M.: Scalable tensor factorizations with missing data. In: Proceedings of the 2010 SIAM International Conference on Data Mining, pp. 701–712 (2010)

    Google Scholar 

  2. Adams, R.P., Dahl, G.E., Murray, I.: Incorporating side information into probabilistic matrix factorization using Gaussian processes. In: Grünwald, P., Spirtes, P. (eds.) Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence, pp. 1–9 (2010)

    Google Scholar 

  3. Cai, D., He, X., Han, J., Huang, T.S.: Graph regularized non-negative matrix factorization for data representation. IEEE Transactions on Pattern Analysis and Machine Intelligence (2010)

    Google Scholar 

  4. Chu, W., Ghahramani, Z.: Probabilistic models for incomplete multi-dimensional arrays. In: Proceedings of the 12th International Conference on Artificial Intelligence and Statistics (2009)

    Google Scholar 

  5. Collins, M., Dasgupta, S., Schapire, R.E.: A generalization of principal components analysis to the exponential family. In: Dietterich, T.G., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems, vol. 14, MIT Press, Cambridge (2002)

    Google Scholar 

  6. Dunlavy, D.M., Kolda, T.G., Acar, E.: Temporal link prediction using matrix and tensor factorizations. ACM Transactions on Knowledge Discovery from Data 5, 10:1–10:27 (2011)

    Google Scholar 

  7. 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 

  8. Hayashi, K., Takenouchi, T., Shibata, T., Kamiya, Y., Kato, D., Kunieda, K., Yamada, K., Ikeda, K.: Exponential family tensor factorization for missing-values prediction and anomaly detection. In: Proceedings of the 10th IEEE International Conference on Data Mining, pp. 216–225 (2010)

    Google Scholar 

  9. Kashima, H., Kato, T., Yamanishi, Y., Sugiyama, M., Tsuda, K.: Link propagation: A fast semi-supervised algorithm for link prediction. In: Proceedings of the 2009 SIAM International Conference on Data Mining (2009)

    Google Scholar 

  10. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Review 51(3), 455–500 (2009)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Kolda, T.G., Sun, J.: Scalable tensor decompositions for multi-aspect data mining. In: Proceedings of the 8th IEEE International Conference on Data Mining, pp. 363–372 (2008)

    Google Scholar 

  12. Li, W.-J., Yeung, D.-Y.: Relation regularized matrix factorization. In: Proceedings of the 21st International Joint Conference on Artificial Intelligence, pp. 1126–1131 (2009)

    Google Scholar 

  13. Lu, Z., Agarwal, D., Dhillon, I.S.: A spatio-temporal approach to collaborative filtering. In: Proceedings of the 3rd ACM Conference on Recommender Systems, pp. 13–20 (2009)

    Google Scholar 

  14. Porteous, I., Asuncion, A., Welling, M.: Bayesian matrix factorization with side information and Dirichlet process mixtures. In: Proceedings of the 24th AAAI Conference on Artificial Intelligence, pp. 563–568 (2010)

    Google Scholar 

  15. Rendle, S., Thieme, L.S.: Pairwise interaction tensor factorization for personalized tag recommendation. In: Proceedings of the 3rd ACM International Conference on Web Search and Data Mining, pp. 81–90 (2010)

    Google Scholar 

  16. Salakhutdinov, R., Mnih, A.: Probabilistic matrix factorization. In: Platt, J.C., Koller, D., Singer, Y., Roweis, S. (eds.) Advances in Neural Information Processing Systems, vol. 20. MIT Press, Cambridge (2008)

    Google Scholar 

  17. Shashua, A., Hazan, T.: Non-negative tensor factorization with applications to statistics and computer vision. In: Proceedings of the 22nd International Conference on Machine Learning, pp. 792–799 (2005)

    Google Scholar 

  18. Srebro, N.: Learning with matrix factorizations. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA (2004)

    Google Scholar 

  19. Tucker, L.: Some mathematical notes on three-mode factor analysis. Psychometrika 31(3), 279–311 (1966)

    MathSciNet  CrossRef  Google Scholar 

  20. Walczak, B.: Dealing with missing data Part I. Chemometrics and Intelligent Laboratory Systems 58(1), 15–27 (2001)

    CrossRef  Google Scholar 

  21. Yu, K., Lafferty, J., Zhu, S., Gong, Y.: Large-scale collaborative prediction using a nonparametric random effects model. In: Proceedings of the 26th International Conference on Machine Learning, pp. 1185–1192 (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Narita, A., Hayashi, K., Tomioka, R., Kashima, H. (2011). Tensor Factorization Using Auxiliary Information. In: Gunopulos, D., Hofmann, T., Malerba, D., Vazirgiannis, M. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2011. Lecture Notes in Computer Science(), vol 6912. Springer, Berlin, Heidelberg.

Download citation

  • DOI:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23782-9

  • Online ISBN: 978-3-642-23783-6

  • eBook Packages: Computer ScienceComputer Science (R0)