Hierarchical nuclear norm penalization for multi-view data

TL;DR

This paper introduces a hierarchical nuclear norm penalty for multi-view data integration, enabling better modeling of partially-shared structures through a convex optimization framework and hierarchical grouping of views.

Contribution

It proposes a novel hierarchical nuclear norm penalty that models partially-shared signals in multi-view data, overcoming limitations of previous methods.

Findings

Outperforms existing methods in simulations

Effectively models partially-shared structures

Demonstrates advantages on genotype-tissue data

Abstract

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.