  Statistical approaches that successfully combine multiple datasets are more
powerful, efficient, and scientifically informative than separate analyses. To
address variation architectures correctly and comprehensively for
high-dimensional data across multiple sample sets (i.e., cohorts), we propose
multiple augmented reduced rank regression (maRRR), a flexible matrix
regression and factorization method to concurrently learn both covariate-driven
and auxiliary structured variation. We consider a structured nuclear norm
objective that is motivated by random matrix theory, in which the regression or
factorization terms may be shared or specific to any number of cohorts. Our
framework subsumes several existing methods, such as reduced rank regression
and unsupervised multi-matrix factorization approaches, and includes a
promising novel approach to regression and factorization of a single dataset
(aRRR) as a special case. Simulations demonstrate substantial gains in power
from combining multiple datasets, and from parsimoniously accounting for all
structured variation. We apply maRRR to gene expression data from multiple
cancer types (i.e., pan-cancer) from TCGA, with somatic mutations as
covariates. The method performs well with respect to prediction and imputation
of held-out data, and provides new insights into mutation-driven and auxiliary
variation that is shared or specific to certain cancer types.
