Matrix and Tensor Decompositions

Advances in high-throughput technologies such as gene and protein expression microarrays in the past decade have made it possible to simultaneously measure the expression levels of thousands of transcripts. This has resulted in large amounts of biological data requiring analysis and interpretation. Many methods for handling such large-scale data have been proposed in the literature. For example, consider a

p

×

n

gene expression matrix

V

consisting of observations on p genes from n samples representing different experimental conditions, phenotypes or time points. One could be interested in identifying clusters of genes with similar expression profiles across sub-groups of samples. Typically, this is accomplished via a decomposition of

V

into two or more matrices where each factored matrix has a distinct physical interpretation. Matrix decompositions have been successfully utilized in a variety of applications in computational biology such as molecular pattern discovery, class comparison, class prediction, functional characterization of genes, cross-platform and cross-species analysis, and biomedical informatics. In this chapter, we focus on available and commonly utilized methods for such matrix decompositions as well as survey other potentially useful methods for analyzing highdimensional data.