Data Mining and Knowledge Discovery

, Volume 29, Issue 5, pp 1343–1373 | Cite as

Clustering Boolean tensors

  • Saskia MetzlerEmail author
  • Pauli Miettinen


Graphs—such as friendship networks—that evolve over time are an example of data that are naturally represented as binary tensors. Similarly to analysing the adjacency matrix of a graph using a matrix factorization, we can analyse the tensor by factorizing it. Unfortunately, tensor factorizations are computationally hard problems, and in particular, are often significantly harder than their matrix counterparts. In case of Boolean tensor factorizations—where the input tensor and all the factors are required to be binary and we use Boolean algebra—much of that hardness comes from the possibility of overlapping components. Yet, in many applications we are perfectly happy to partition at least one of the modes. For instance, in the aforementioned time-evolving friendship networks, groups of friends might be overlapping, but the time points at which the network was captured are always distinct. In this paper we investigate what consequences this partitioning has on the computational complexity of the Boolean tensor factorizations and present a new algorithm for the resulting clustering problem. This algorithm can alternatively be seen as a particularly regularized clustering algorithm that can handle extremely high-dimensional observations. We analyse our algorithm with the goal of maximizing the similarity and argue that this is more meaningful than minimizing the dissimilarity. As a by-product we obtain a PTAS and an efficient 0.828-approximation algorithm for rank-1 binary factorizations. Our algorithm for Boolean tensor clustering achieves high scalability, high similarity, and good generalization to unseen data with both synthetic and real-world data sets.


Tensors Clustering Boolean algebra Approximation Decomposition 


  1. Alon N, Sudakov B (1999) On two segmentation problems. J Algorithm 33:173–184MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bělohlávek R, Glodeanu C, Vychodil V (2012) Optimal factorization of three-way binary data using triadic concepts. Order 30(2):437–454CrossRefGoogle Scholar
  3. Cantador I, Brusilovsky P, Kuflik T (2011) 2nd Workshop on Information Heterogeneity and Fusion in Recommender Systems (HetRec ’11). In: 5th ACM Conference on Recommender Systems (RecSys’11)Google Scholar
  4. Carroll JD, Chang JJ (1970) Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika 35(3):283–319CrossRefzbMATHGoogle Scholar
  5. Cerf L, Besson J, Robardet C, Boulicaut JF (2009) Closed patterns meet n-ary relations. ACM Trans Knowl Discov Data 3(1):1CrossRefGoogle Scholar
  6. Cerf L, Besson J, Nguyen KNT, Boulicaut JF (2013) Closed and noise-tolerant patterns in n-ary relations. Data Min Knowl Discov 26(3):574–619MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chi EC, Kolda TG (2012) On tensors, sparsity, and nonnegative factorizations. SIAM J Matrix Anal Appl 33(4):1272–1299MathSciNetCrossRefzbMATHGoogle Scholar
  8. Dagum L, Menon R (1998) OpenMP: an industry standard API for shared-memory programming. IEEE Comput Sci Eng Mag 5(1):46–55CrossRefGoogle Scholar
  9. Erdős D, Miettinen P (2013a) Discovering facts with boolean tensor tucker decomposition. In: 22nd ACM International Conference on Information & Knowledge Management (CIKM ’13), pp 1569–1572Google Scholar
  10. Erdős D, Miettinen P (2013b) Walk’n’Merge: a scalable algorithm for Boolean tensor factorization. In: 13th IEEE International Conference on Data Mining (ICDM ’13), pp 1037–1042Google Scholar
  11. Harshman RA (1970) Foundations of the PARAFAC procedure: models and conditions for an “explanatory” multimodal factor analysis. Tech. Rep. 16, UCLA Working Papers in PhoneticsGoogle Scholar
  12. Huang H, Ding C, Luo D, Li T (2008) Simultaneous tensor subspace selection and clustering: the equivalence of high order SVD and k-means clustering. In: 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD ’08), pp 327–335Google Scholar
  13. Ignatov DI, Kuznetsov SO, Magizov RA, Zhukov LE (2011) From triconcepts to triclusters. In: 13th International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing (RSFDGrC ’11), pp 257–264Google Scholar
  14. Jegelka S, Sra S, Banerjee A (2009) Approximation algorithms for tensor clustering. In: International Conference on Algorithmic Learning Theory (ALT ’09), pp 368–383Google Scholar
  15. Jiang P (2014) Pattern extraction and clustering for high-dimensional discrete data. PhD thesis, University of Illinois at Urbana-ChampaignGoogle Scholar
  16. Kim M, Candan KS (2011) Approximate tensor decomposition within a tensor-relational algebraic framework. In: 20th ACM International Conference on Information & Knowledge Management (CIKM ’11), pp 1737–1742Google Scholar
  17. Kim M, Candan KS (2012) Decomposition-by-normalization (DBN): leveraging approximate functional dependencies for efficient tensor decomposition. In: 21st ACM International Conference on Information & Knowledge Management (CIKM ’12), pp 355–364Google Scholar
  18. Kim M, Candan KS (2014) Pushing-down tensor decompositions over unions to promote reuse of materialized decompositions. In: European Conference on Machine Learning and Knowledge Discovery in Databases (ECML PKDD ’14), pp 688–704Google Scholar
  19. Kleinberg J, Papadimitriou C, Raghavan P (1998) A microeconomic view of data mining. Data Min Knowl Discov 2(4):311–324CrossRefGoogle Scholar
  20. Kleinberg JM, Papadimitriou CH, Raghavan P (2004) Segmentation problems. J ACM 51(2):263–280MathSciNetCrossRefGoogle Scholar
  21. Kolda TG, Bader BW (2009) Tensor decompositions and applications. SIAM Rev 51(3):455–500MathSciNetCrossRefzbMATHGoogle Scholar
  22. Leenen I, Van Mechelen I, De Boeck P, Rosenberg S (1999) INDCLAS: a three-way hierarchical classes model. Psychometrika 64(1):9–24CrossRefzbMATHGoogle Scholar
  23. Liu X, De Lathauwer L, Janssens F, De Moor B (2010) Hybrid clustering of multiple information sources via HOSVD. In: 7th International Conference on Advances in Neural Networks—Part II (ISNN ’10), pp 337–345Google Scholar
  24. Miettinen P (2009) Matrix Decomposition methods for data mining: computational complexity and algorithms. PhD thesis, Department of Computer Science, University of HelsinkiGoogle Scholar
  25. Miettinen P (2010) Sparse Boolean matrix factorizations. In: 10th IEEE International Conference on Data Mining (ICDM ’10), pp 935–940Google Scholar
  26. Miettinen P (2011) Boolean tensor factorizations. In: 11th IEEE International Conference on Data Mining (ICDM ’11), pp 447–456Google Scholar
  27. Miettinen P, Vreeken J (2014) MDL4BMF: minimum description length for Boolean matrix factorization. ACM Trans Knowl Discov Data 8(4):18CrossRefGoogle Scholar
  28. Miettinen P, Mielikäinen T, Gionis A, Das G, Mannila H (2008) The discrete basis problem. IEEE Trans Knowl Data Eng 20(10):1348–1362CrossRefGoogle Scholar
  29. Papadimitriou CH, Steiglitz K (1998) Combinatorial optimization: algorithms and complexity. Dover Publications, MineolazbMATHGoogle Scholar
  30. Papalexakis EE, Faloutsos C, Sidiropoulos ND (2012) ParCube: sparse parallelizable tensor decompositions. In: European Conference on Machine Learning and Knowledge Discovery in Databases (ECML PKDD ’12), pp 521–536Google Scholar
  31. Papalexakis EE, Sidiropoulos N, Bro R (2013) From K-means to higher-way co-clustering: multilinear decomposition with sparse latent factors. IEEE Trans Signal Process 61(2):493–506CrossRefGoogle Scholar
  32. Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471CrossRefzbMATHGoogle Scholar
  33. Seppänen JK (2005) Upper bound for the approximation ratio of a class of hypercube segmentation algorithms. Inform Process Lett 93(3):139–141MathSciNetCrossRefzbMATHGoogle Scholar
  34. Suchanek FM, Kasneci G, Weikum G (2007) Yago: a core of semantic knowledge. In: 16th International Conference on World Wide Web (WWW ’07), pp 697–706Google Scholar
  35. Tucker LR (1966) Some mathematical notes on three-mode factor analysis. Psychometrika 31(3):279–311MathSciNetCrossRefGoogle Scholar
  36. Viswanath B, Mislove A, Cha M, Gummadi KP (2009) On the evolution of user interaction in Facebook. In: 2nd ACM Workshop on Online Social Networks (WOSN ’09), pp 37–42Google Scholar
  37. Yates A, Etzioni O (2009) Unsupervised methods for determining object and relation synonyms on the web. J Artif Intell Res 34:255–296zbMATHGoogle Scholar
  38. Zhao L, Zaki MJ (2005) TRICLUSTER: an effective algorithm for mining coherent clusters in 3D microarray data. In: ACM SIGMOD International Conference on Management of Data (SIGMOD ’05), pp 694–705Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations