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Several modern applications require the integration of multiple large datamatrices that have shared rows and/or columns. For example, cancer studies thatintegrate multiple omics platforms across multiple types of cancer, pan-omicspan-cancer analysis, have extended our knowledge of molecular heterogenitybeyond what was observed in single tumor and single platform studies. However,these studies have been limited by available statistical methodology. Wepropose a flexible approach to the simultaneous factorization and decompositionof variation across such bidimensionally linked matrices, BIDIFAC+. Thisdecomposes variation into a series of low-rank components that may be sharedacross 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 anddecomposition of linked matrices, which has primarily focused on multiplematrices that are linked in one dimension (rows or columns) only. Our objectivefunction extends nuclear norm penalization, is motivated by random matrixtheory, gives an identifiable decomposition under relatively mild conditions,and can be shown to give the mode of a Bayesian posterior distribution. Weapply BIDIFAC+ to pan-omics pan-cancer data from TCGA, identifying shared andspecific modes of variability across 4 different omics platforms and 29different cancer types.

Bidimensional linked matrix factorization for pan-omics pan-cancer analysis