  Data integration, or the strategic analysis of multiple sources of data
simultaneously, can often lead to discoveries that may be hidden in
individualistic analyses of a single data source. We develop a new unsupervised
data integration method named Integrated Principal Components Analysis (iPCA),
which is a model-based generalization of PCA and serves as a practical tool to
find and visualize common patterns that occur in multiple data sets. The key
idea driving iPCA is the matrix-variate normal model, whose Kronecker product
covariance structure captures both individual patterns within each data set and
joint patterns shared by multiple data sets. Building upon this model, we
develop several penalized (sparse and non-sparse) covariance estimators for
iPCA, and using geodesic convexity, we prove that our non-sparse iPCA estimator
converges to the global solution of a non-convex problem. We also demonstrate
the practical advantages of iPCA through extensive simulations and a case study
application to integrative genomics for Alzheimer's disease. In particular, we
show that the joint patterns extracted via iPCA are highly predictive of a
patient's cognition and Alzheimer's diagnosis.
