Genevera Allen
Baylor College of Medicine & Rice University

A Generalized Least Squares Matrix Decomposition

Variables in high-dimensional data sets common in neuroimaging and genomics often exhibit complex dependencies. These relationships, due to spatio-temporal processes, network structures, or latent variables, for example, are often ignored by conventional multivariate analysis techniques. We propose a generalization of the singular value decomposition (SVD) that is appropriate for transposable matrix data, or data in which neither the rows nor the columns can be considered independent instances. Our decomposition, entitled the Generalized least squares Matrix Decomposition (GMD), finds the best low rank approximation to the data with respect to a transposable quadratic norm. By adding penalties to the factors, we introduce the Generalized Penalized Matrix Factorization (GPMF). We show that the GMD can be used for generalized PCA and the GPMF for sparse GPCA and functional GPCA. We also outline extensions of our methodology for statistical applications such as canonical correlation analysis, non-negative matrix factorization, and linear discriminant analysis. Through simulations and examples we demonstrate the utility of the GMD and GPMF for dimension reduction, sparse and functional signal recovery, and feature selection with high-dimensional transposable data. Real data examples on functional MRIs are given.