  Data for several applications in diverse fields can be represented as
multiple matrices that are linked across rows or columns. This is particularly
common in molecular biomedical research, in which multiple molecular "omics"
technologies may capture different feature sets (e.g., corresponding to rows in
a matrix) and/or different sample populations (corresponding to columns). This
has motivated a large body of work on integrative matrix factorization
approaches that identify and decompose low-dimensional signal that is shared
across multiple matrices or specific to a given matrix. We propose an empirical
variational Bayesian approach to this problem that has several advantages over
existing techniques, including the flexibility to accommodate shared signal
over any number of row or column sets (i.e., bidimensional integration), an
intuitive model-based objective function that yields appropriate shrinkage for
the inferred signals, and a relatively efficient estimation algorithm with no
tuning parameters. A general result establishes conditions for the uniqueness
of the underlying decomposition for a broad family of methods that includes the
proposed approach. For scenarios with missing data, we describe an associated
iterative imputation approach that is novel for the single-matrix context and a
powerful approach for "blockwise" imputation (in which an entire row or column
is missing) in various linked matrix contexts. Extensive simulations show that
the method performs very well under different scenarios with respect to
recovering underlying low-rank signal, accurately decomposing shared and
specific signals, and accurately imputing missing data. The approach is applied
to gene expression and miRNA data from breast cancer tissue and normal breast
tissue, for which it gives an informative decomposition of variation and
outperforms alternative strategies for missing data imputation.
