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Recovering all generalized order-preserving submatrices: new exact formulations and algorithms

Author

Listed:
  • Andrew C. Trapp

    (Worcester Polytechnic Institute)

  • Chao Li

    (Worcester Polytechnic Institute)

  • Patrick Flaherty

    (University of Massachusetts)

Abstract

Cluster analysis of gene expression data is a popular and successful way of elucidating underlying biological processes. Typically, cluster analysis methods seek to group genes that are differentially expressed across experimental conditions. However, real biological processes often involve only a subset of genes and are activated in only a subset of environmental or temporal conditions. To address this limitation, Ben-Dor et al. (J Comput Biol 10(3–4):373–384, 2003) developed an approach to identify order-preserving submatrices (OPSMs) in which the expression levels of included genes induce the sample linear ordering of experiments. In addition to gene expression analysis, OPSMs have application to recommender systems and target marketing. While the problem of finding the largest OPSM is $${{\mathscr {N}}}{{\mathscr {P}}}$$ N P -hard, there have been significant advances in both exact and approximate algorithms in recent years. Building upon these developments, we provide two exact mathematical programming formulations that generalize the OPSM formulation by allowing for the reverse linear ordering, known as the generalized OPSM pattern, or GOPSM. Our formulations incorporate a constraint that provides a margin of safety against detecting spurious GOPSMs. Finally, we provide two novel algorithms to recover, for any given level of significance, all GOPSMs from a given data matrix, by iteratively solving mathematical programming formulations to global optimality. We demonstrate the computational performance and accuracy of our algorithms on real gene expression data sets showing the capability of our developments.

Suggested Citation

  • Andrew C. Trapp & Chao Li & Patrick Flaherty, 2018. "Recovering all generalized order-preserving submatrices: new exact formulations and algorithms," Annals of Operations Research, Springer, vol. 263(1), pages 385-404, April.
  • Handle: RePEc:spr:annopr:v:263:y:2018:i:1:d:10.1007_s10479-016-2173-9
    DOI: 10.1007/s10479-016-2173-9
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    References listed on IDEAS

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    1. Andrew C. Trapp & Oleg A. Prokopyev, 2010. "Solving the Order-Preserving Submatrix Problem via Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 387-400, August.
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    Cited by:

    1. Xinghua Fang & Jian Zhou & Hongya Zhao & Yizeng Chen, 2022. "A biclustering-based heterogeneous customer requirement determination method from customer participation in product development," Annals of Operations Research, Springer, vol. 309(2), pages 817-835, February.

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