  The increased availability of the multi-view data (data on the same samples
from multiple sources) has led to strong interest in models based on low-rank
matrix factorizations. These models represent each data view via shared and
individual components, and have been successfully applied for exploratory
dimension reduction, association analysis between the views, and further
learning tasks such as consensus clustering. Despite these advances, there
remain significant challenges in modeling partially-shared components, and
identifying the number of components of each type
(shared/partially-shared/individual). In this work, we formulate a novel linked
component model that directly incorporates partially-shared structures. We call
this model SLIDE for Structural Learning and Integrative DEcomposition of
multi-view data. We prove the existence of SLIDE decomposition and explicitly
characterize the identifiability conditions. The proposed model fitting and
selection techniques allow for joint identification of the number of components
of each type, in contrast to existing sequential approaches. In our empirical
studies, SLIDE demonstrates excellent performance in both signal estimation and
component selection. We further illustrate the methodology on the breast cancer
data from The Cancer Genome Atlas repository.
