Rank Matrix Factorisation

  • Thanh Le Van
  • Matthijs van Leeuwen
  • Siegfried Nijssen
  • Luc De Raedt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9077)

Abstract

We introduce the problem of rank matrix factorisation (RMF). That is, we consider the decomposition of a rank matrix, in which each row is a (partial or complete) ranking of all columns. Rank matrices naturally appear in many applications of interest, such as sports competitions. Summarising such a rank matrix by two smaller matrices, in which one contains partial rankings that can be interpreted as local patterns, is therefore an important problem.

After introducing the general problem, we consider a specific instance called Sparse RMF, in which we enforce the rank profiles to be sparse, i.e., to contain many zeroes. We propose a greedy algorithm for this problem based on integer linear programming. Experiments on both synthetic and real data demonstrate the potential of rank matrix factorisation.

Keywords

Matrix factorisation Rank data Integer linear programming 

References

  1. 1.
    Le Van, T., van Leeuwen, M., Nijssen, S., Fierro, A.C., Marchal, K., De Raedt, L.: Ranked tiling. In: Calders, T., Esposito, F., Hüllermeier, E., Meo, R. (eds.) ECML PKDD 2014, Part II. LNCS, vol. 8725, pp. 98–113. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  2. 2.
    Henzgen, S., Hüllermeier, E.: Mining rank data. In: Džeroski, S., Panov, P., Kocev, D., Todorovski, L. (eds.) DS 2014. LNCS (LNAI), vol. 8777, pp. 123–134. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  3. 3.
    Paatero, P., Tapper, U.: Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values. Environmetrics 5(2), 111–126 (1994)CrossRefGoogle Scholar
  4. 4.
    Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401(6755), 788–791 (1999)CrossRefGoogle Scholar
  5. 5.
    Monson, S., Pullman, N., Rees, R.: A survey of clique and biclique coverings and factorizations of (0, 1)-matrices. Bull. Inst. Combinatorics and Its Applications 14, 17–86 (1995)MATHMathSciNetGoogle Scholar
  6. 6.
    Miettinen, P., Mielikainen, T., Gionis, A., Das, G., Mannila, H.: The discrete basis problem. IEEE Transactions on Knowledge and Data Engineering 20(10), 1348–1362 (2008)CrossRefGoogle Scholar
  7. 7.
    Lust, T., Teghem, J.: Multiobjective decomposition of positive integer matrix: application to radiotherapy. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 335–349. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  8. 8.
    Marden, J.I.: Analyzing and Modeling Rank Data. Chapman & Hall (1995)Google Scholar
  9. 9.
    Madeira, S.C., Oliveira, A.L.: Biclustering algorithms for biological data analysis: a survey. IEEE/ACM Transactions on Computational Biology and Bioinformatics 1(1), 24–45 (2004)CrossRefGoogle Scholar
  10. 10.
    Cheng, Y., Church, G.M.: Biclustering of expression data. In: Proc. of the 8th International Conference on Intelligent Systems for Molecular Biology, vol. 8, pp. 93–103 (2000)Google Scholar
  11. 11.
    Kluger, Y., Basri, R., Chang, J.T., Gerstein, M.: Spectral Biclustering of Microarray Data : Coclustering Genes and Conditions. Genome Research 13, 703–716 (2003)CrossRefGoogle Scholar
  12. 12.
    Turner, H., Bailey, T., Krzanowski, W.: Improved biclustering of microarray data demonstrated through systematic performance tests. Computational Statistics & Data Analysis 48(2), 235–254 (2005)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Hochreiter, S., Bodenhofer, U., Heusel, M., Mayr, A., Mitterecker, A., Kasim, A., Khamiakova, T., Van Sanden, S., Lin, D., Talloen, W., Bijnens, L., Göhlmann, H.W.H., Shkedy, Z., Clevert, D.A.: FABIA: factor analysis for bicluster acquisition. Bioinformatics 26(12), 1520–1527 (2010)CrossRefGoogle Scholar
  14. 14.
    Tanay, A., Sharan, R., Shamir, R.: Discovering statistically significant biclusters in gene expression data. Bioinformatics 18(Suppl. 1), S136–S144 (2002)CrossRefGoogle Scholar
  15. 15.
    Ihmels, J., Friedlander, G., Bergmann, S., Sarig, O., Ziv, Y., Barkai, N.: Revealing modular organization in the yeast transcriptional network. Nature Genetics 31(4), 370–377 (2002)Google Scholar
  16. 16.
    Truong, D.T., Battiti, R., Brunato, M.: Discovering non-redundant overlapping biclusters on gene expression data. In: ICDM 2013, pp. 747–756. IEEE (2013)Google Scholar
  17. 17.
    Deng, K., Han, S., Li, K.J., Liu, J.S.: Bayesian Aggregation of Order-Based Rank Data. Journal of the American Statistical Association 109(507), 1023–1039 (2014)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kamishima, T.: Nantonac collaborative filtering: recommendation based on order responses. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2003, New York, NY, USA, pp. 583–588. ACM (2003)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Thanh Le Van
    • 1
  • Matthijs van Leeuwen
    • 1
  • Siegfried Nijssen
    • 1
    • 2
  • Luc De Raedt
    • 1
  1. 1.Department of Computer ScienceKU LeuvenLeuvenBelgium
  2. 2.Leiden Institute for Advanced Computer ScienceUniversiteit LeidenLeidenThe Netherlands

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