  The prevalence of data collected on the same set of samples from multiple
sources (i.e., multi-view data) has prompted significant development of data
integration methods based on low-rank matrix factorizations. These methods
decompose signal matrices from each view into the sum of shared and individual
structures, which are further used for dimension reduction, exploratory
analyses, and quantifying associations across views. However, existing methods
have limitations in modeling partially-shared structures due to either too
restrictive models, or restrictive identifiability conditions. To address these
challenges, we formulate a new model for partially-shared signals based on
grouping the views into so-called hierarchical levels. The proposed hierarchy
leads us to introduce a new penalty, hierarchical nuclear norm (HNN), for
signal estimation. In contrast to existing methods, HNN penalization avoids
scores and loadings factorization of the signals and leads to a convex
optimization problem, which we solve using a dual forward-backward algorithm.
We propose a simple refitting procedure to adjust the penalization bias and
develop an adapted version of bi-cross-validation for selecting tuning
parameters. Extensive simulation studies and analysis of the genotype-tissue
expression data demonstrate the advantages of our method over existing
alternatives.
