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Heuristics for Exact Nonnegative Matrix Factorization

Author

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  • VANDAELE, Arnaud
  • GILLIS, Nicolas
  • GLINEUR, François
  • TUYTTENS, Daniel

Abstract

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an m-by-n nonnegative matrix X and a factorization rank r, find, if possible, an m-by-r nonnegative matrix W and an r-by-n nonnegative matrix H such that X = WH. In this paper, we propose two heuristics for exact NMF, one inspired from simulated annealing and the other from the greedy randomized adaptive search procedure. We show that these two heuristics are able to compute exact nonnegative factorizations for several classes of nonnegative matrices (namely, linear Euclidean distance matrices, slack matrices, unique-disjointness matrices, and randomly generated matrices) and as such demonstrate their superiority over standard multi-start strategies. We also consider a hybridization between these two heuristics that allows us to combine the advantages of both methods. Finally, we discuss the use of these heuristics to gain insight on the behavior of the nonnegative rank, i.e., the minimum factorization rank such that an exact NMF exists. In particular, we disprove a conjecture on the nonnegative rank of a Kronecker product, propose a new upper bound on the extension complexity of generic n-gons and conjecture the exact value of (i) the extension complexity of regular n-gons and (ii) the nonnegative rank of a submatrix of the slack matrix of the correlation polytope.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • VANDAELE, Arnaud & GILLIS, Nicolas & GLINEUR, François & TUYTTENS, Daniel, 2016. "Heuristics for Exact Nonnegative Matrix Factorization," LIDAM Reprints CORE 2737, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvrp:2737
    Note: In : Journal of Global Optimization, 65(2) 2016, p. 369-400
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    References listed on IDEAS

    as
    1. João Gouveia & Pablo A. Parrilo & Rekha R. Thomas, 2013. "Lifts of Convex Sets and Cone Factorizations," Mathematics of Operations Research, INFORMS, vol. 38(2), pages 248-264, May.
    2. GILLIS, Nicolas & GLINEUR, François, 2011. "Accelerated multiplicative updates and hierarchical als algorithms for nonnegative matrix factorization," LIDAM Discussion Papers CORE 2011030, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. GILLIS, Nicolas & GLINEUR, François, 2010. "On the geometric interpretation of the nonnegative rank," LIDAM Discussion Papers CORE 2010051, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Andreas Janecek & Ying Tan, 2011. "Swarm Intelligence for Non-Negative Matrix Factorization," International Journal of Swarm Intelligence Research (IJSIR), IGI Global, vol. 2(4), pages 12-34, October.
    5. Jingu Kim & Yunlong He & Haesun Park, 2014. "Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework," Journal of Global Optimization, Springer, vol. 58(2), pages 285-319, February.
    6. repec:cor:louvrp:-2187 is not listed on IDEAS
    7. Pirlot, Marc, 1996. "General local search methods," European Journal of Operational Research, Elsevier, vol. 92(3), pages 493-511, August.
    8. Norikazu Takahashi & Ryota Hibi, 2014. "Global convergence of modified multiplicative updates for nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 57(2), pages 417-440, March.
    9. repec:cor:louvrp:-2439 is not listed on IDEAS
    10. GILLIS, Nicolas & GLINEUR, François, 2009. "Using underapproximations for sparse nonnegative matrix factorization," LIDAM Discussion Papers CORE 2009006, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    11. Aharon Ben-Tal & Arkadi Nemirovski, 2001. "On Polyhedral Approximations of the Second-Order Cone," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 193-205, May.
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    Cited by:

    1. Veit Elser, 2017. "Matrix product constraints by projection methods," Journal of Global Optimization, Springer, vol. 68(2), pages 329-355, June.
    2. Yukihiro Nishimura & Pierre Pestieau, 2016. "Efficient taxation with differential risks of dependence and mortality," Economics Bulletin, AccessEcon, vol. 36(1), pages 52-57.
    3. Melisew Tefera Belachew & Nicolas Gillis, 2017. "Solving the Maximum Clique Problem with Symmetric Rank-One Non-negative Matrix Approximation," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 279-296, April.
    4. Arnaud Vandaele & François Glineur & Nicolas Gillis, 2018. "Algorithms for positive semidefinite factorization," Computational Optimization and Applications, Springer, vol. 71(1), pages 193-219, September.

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