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On the Complexity of Robust PCA and ℓ 1 -Norm Low-Rank Matrix Approximation

Author

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  • Nicolas Gillis

    (Department of Mathematics and Operational Research, University of Mons, 7000 Mons, Belgium)

  • Stephen A. Vavasis

    (Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

Abstract

The low-rank matrix approximation problem with respect to the component-wise ℓ 1 -norm ( ℓ 1 -LRA), which is closely related to robust principal component analysis (PCA), has become a very popular tool in data mining and machine learning. Robust PCA aims to recover a low-rank matrix that was perturbed with sparse noise, with applications for example in foreground-background video separation. Although ℓ 1 -LRA is strongly believed to be NP-hard, there is, to our knowledge, no formal proof of this fact. In this paper, we prove that ℓ 1 -LRA is NP-hard, already in the rank-one case, using a reduction from MAX CUT. Our derivations draw interesting connections between ℓ 1 -LRA and several other well-known problems, i.e., robust PCA, ℓ 0 -LRA, binary matrix factorization, a particular densest bipartite subgraph problem, the computation of the cut norm of {−1, + 1} matrices, and the discrete basis problem, all of which we prove to be NP-hard.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormoor:v:43:y:2018:i:4:p:1072-1084
    DOI: 10.1287/moor.2017.0895
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    References listed on IDEAS

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    1. 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).
    2. GILLIS, Nicolas & GLINEUR, François, 2010. "Low-rank matrix approximation with weights or missing data is NP-hard," LIDAM Discussion Papers CORE 2010075, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Ali Hamzenejad & Saeid Jafarzadeh Ghoushchi & Vahid Baradaran & Abbas Mardani, 2020. "A Robust Algorithm for Classification and Diagnosis of Brain Disease Using Local Linear Approximation and Generalized Autoregressive Conditional Heteroscedasticity Model," Mathematics, MDPI, vol. 8(8), pages 1-19, August.

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