Tensor-on-tensor regression

We propose a framework for the linear prediction of a multi-way array (i.e., a tensor) from another multi-way array of arbitrary dimension, using the contracted tensor product. This framework generalizes several existing approaches, including methods to predict a scalar outcome from a tensor, a matrix from a matrix, or a tensor from a scalar. We describe an approach that exploits the multiway structure of both the predictors and the outcomes by restricting the coefficients to have reduced CP-rank. We propose a general and efficient algorithm for penalized least-squares estimation, which allows for a ridge (L_2) penalty on the coefficients. The objective is shown to give the mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for inference. We illustrate the approach with an application to facial image data. An R package is available at https://github.com/lockEF/MultiwayRegression .Comment: 33 pages, 3 figure

Prediction with Dimension Reduction of Multiple Molecular Data Sources for Patient Survival

Predictive modeling from high-dimensional genomic data is often preceded by a dimension reduction step, such as principal components analysis (PCA). However, the application of PCA is not straightforward for multi-source data, wherein multiple sources of 'omics data measure different but related biological components. In this article we utilize recent advances in the dimension reduction of multi-source data for predictive modeling. In particular, we apply exploratory results from Joint and Individual Variation Explained (JIVE), an extension of PCA for multi-source data, for prediction of differing response types. We conduct illustrative simulations to illustrate the practical advantages and interpretability of our approach. As an application example we consider predicting survival for Glioblastoma Multiforme (GBM) patients from three data sources measuring mRNA expression, miRNA expression, and DNA methylation. We also introduce a method to estimate JIVE scores for new samples that were not used in the initial dimension reduction, and study its theoretical properties; this method is implemented in the R package R.JIVE on CRAN, in the function 'jive.predict'.Comment: 11 pages, 9 figure