Integrative Multivariate Learning via Composite Low-Rank Decompositions

We develop novel composite low-rank methods to achieve integrative learning in multivariate linear regression, where both the multivariate responses and predictors can be of high dimensionality and in different data forms. We first consider a regression with multi-view feature sets where only a few views are relevant to prediction and the predictors within each relevant view contribute to the prediction collectively rather than sparsely. To tackle this problem, we propose an integrative reduced-rank regression (iRRR) where each view has its own low-rank coefficient matrix, to conduct view selection and within-view latent feature extraction in a supervised fashion. In addition, to assess the significance of each view in iRRR model, we propose a scaled approach for model estimation and develop a hypothesis testing procedure through de-biasing. Next, to facilitate integrative multi-view learning with grouped sub-compositional predictors, we incorporate the view-specific low-rank structure into a newly proposed multivariate log-contrast model to enable sub-composition selection and latent principal compositional factor extraction. Finally, we propose a nested reduced-rank regression (NRRR) approach to relate multivariate functional responses and predictors. The nested low-rank structure is imposed on the functional regression surfaces to simultaneously identify latent principal functional responses/predictors and control the complexity and smoothness of the association between them. Efficient computational algorithms are developed for these methods, and their theoretical properties are investigated. We apply the proposed methods to multiple applications including the longitudinal study of aging, the preterm infant study and the electricity demand prediction.