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

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 ${{\mathscr {N}}}{{\mathscr {P}}}$-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.

Appendix: Additional algorithmic and computational details

Tables 4 and 5 below identify, for a given column level $\gamma$, the corresponding minimum number of rows $\rho _{\gamma }^{\alpha }$ necessary for a GOPSM to meet the statistical significance threshold for level $\alpha$ as motivated in Ben-Dor et al. (2003). These values are computed via (13), and are used in the construction of constraints (14) and (15). We detail these right-hand side values for both the Cooper promoter (Table 4) and the Spellman yeast (Table 5) data sets.