  Modern biomedical studies often collect multi-view data, that is, multiple
types of data measured on the same set of objects. A popular model in
high-dimensional multi-view data analysis is to decompose each view's data
matrix into a low-rank common-source matrix generated by latent factors common
across all data views, a low-rank distinctive-source matrix corresponding to
each view, and an additive noise matrix. We propose a novel decomposition
method for this model, called decomposition-based generalized canonical
correlation analysis (D-GCCA). The D-GCCA rigorously defines the decomposition
on the L2 space of random variables in contrast to the Euclidean dot product
space used by most existing methods, thereby being able to provide the
estimation consistency for the low-rank matrix recovery. Moreover, to well
calibrate common latent factors, we impose a desirable orthogonality constraint
on distinctive latent factors. Existing methods, however, inadequately consider
such orthogonality and may thus suffer from substantial loss of undetected
common-source variation. Our D-GCCA takes one step further than generalized
canonical correlation analysis by separating common and distinctive components
among canonical variables, while enjoying an appealing interpretation from the
perspective of principal component analysis. Furthermore, we propose to use the
variable-level proportion of signal variance explained by common or distinctive
latent factors for selecting the variables most influenced. Consistent
estimators of our D-GCCA method are established with good finite-sample
numerical performance, and have closed-form expressions leading to efficient
computation especially for large-scale data. The superiority of D-GCCA over
state-of-the-art methods is also corroborated in simulations and real-world
data examples.
