D-GCCA: Decomposition-based Generalized Canonical Correlation Analysis for Multiple High-dimensional Datasets

Such studies include The Cancer Genome Atlas (TCGA; Hoadley et al., 2018) with multi-platform genomic data for tumor samples, and Human Connectome Project (HCP; Van Essen et al., 2013) with multi-modal brain images of healthy adults, among many others (Crawford et al., 2016; Jensen et al., 2017). The use of multiple data types can allow us to enhance understanding the etiology of many complex diseases, such as cancers (Ciriello et al., 2015; Campbell et al., 2018) and neurodegenerative diseases (Weiner et al., 2013; Saeed et al., 2017). Researchers hence have became highly interested in studying the shared information and individual features across multi-type datasets through separating their common and distinctive variation structures (van der Kloet et al., 2016; Smilde et al., 2017; Li et al., 2018). Let Y k R p k n be the k -th row-mean centered dataset obtained on a common set of n objects for k 1,...,K, where p k is the number of variables for the k -th dataset. One popular approach for disentangling their common and distinctive variation structures is to decompose each data matrix into Y k X k E k C k D k E k for k 1,...,K, (1) where { X k} K k 1 are low-rank signal matrices with { E k} K k 1 being additive noise matrices, { C k} K k 1 are low-rank common-variation matrices that represent the signal data coming from the common mechanism shared across all datasets, and { D k} K k 1are low-rank distinctive-variation matrices each from the distinctive mechanism of each single dataset that is not shared by all.