  Several modern applications require the integration of multiple large data
matrices that have shared rows and/or columns. For example, cancer studies that
integrate multiple omics platforms across multiple types of cancer, pan-omics
pan-cancer analysis, have extended our knowledge of molecular heterogenity
beyond what was observed in single tumor and single platform studies. However,
these studies have been limited by available statistical methodology. We
propose a flexible approach to the simultaneous factorization and decomposition
of variation across such bidimensionally linked matrices, BIDIFAC+. This
decomposes variation into a series of low-rank components that may be shared
across any number of row sets (e.g., omics platforms) or column sets (e.g.,
cancer types). This builds on a growing literature for the factorization and
decomposition of linked matrices, which has primarily focused on multiple
matrices that are linked in one dimension (rows or columns) only. Our objective
function extends nuclear norm penalization, is motivated by random matrix
theory, gives an identifiable decomposition under relatively mild conditions,
and can be shown to give the mode of a Bayesian posterior distribution. We
apply BIDIFAC+ to pan-omics pan-cancer data from TCGA, identifying shared and
specific modes of variability across 4 different omics platforms and 29
different cancer types.
