Hierarchical nuclear norm penalization for multi‐view data integration

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 propose a new formulation for signal structures that include partially‐shared signals based on grouping the views into so‐called hierarchical levels with identifiable guarantees under suitable conditions. 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.