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Using underapproximations for sparse nonnegative matrix factorization

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  • GILLIS, Nicolas
  • Glineur, François

Abstract

Nonnegative Matrix Factorization (NMF) has gathered a lot of attention in the last decade and has been successfully applied in numerous applications. It consists in the factorization of a nonnegative matrix by the product of two low-rank nonnegative matrices:. MªVW. In this paper, we attempt to solve NMF problems in a recursive way. In order to do that, we introduce a new variant called Nonnegative Matrix Underapproximation (NMU) by adding the upper bound constraint VW£M. Besides enabling a recursive procedure for NMF, these inequalities make NMU particularly well suited to achieve a sparse representation, improving the part-based decomposition. Although NMU is NP-hard (which we prove using its equivalence with the maximum edge biclique problem in bipartite graphs), we present two approaches to solve it: a method based on convex reformulations and a method based on Lagrangian relaxation. Finally, we provide some encouraging numerical results for image processing applications.
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Suggested Citation

  • GILLIS, Nicolas & Glineur, François, 2010. "Using underapproximations for sparse nonnegative matrix factorization," LIDAM Reprints CORE 2187, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:2187
    DOI: 10.1016/j.patcog.2009.11.013
    Note: In : Pattern Recognition, 43, 1676-1687, 2010
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    Cited by:

    1. GILLIS, Nicolas & GLINEUR, François, 2008. "Nonnegative factorization and the maximum edge biclique problem," LIDAM Discussion Papers CORE 2008064, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Gillis, Nicolas & Glineur, François & Tuyttens, Daniel & Vandaele, Arnaud, 2015. "Heuristics for exact nonnegative matrix factorization," LIDAM Discussion Papers CORE 2015006, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Nicolas Gillis & Stephen A. Vavasis, 2018. "On the Complexity of Robust PCA and ℓ 1 -Norm Low-Rank Matrix Approximation," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1072-1084, November.
    4. 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.

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