A Generalized Least Squares Matrix Decomposition with an Application to Neuroimaging

• Activity Number: 227
• Type: Topic Contributed
• Date/Time: Monday, August 1, 2011 : 2:00 PM to 3:50 PM
• Sponsor: Biometrics Section
• Author(s): Genevera I. Allen*+ and Logan Grosenick and Jonathan Taylor
• Companies: Baylor College of Medicine/Rice University and Stanford University and Stanford University
• Keywords: matrix decomposition ; principal components analysis ; sparse principal components analysis ; functional principal components anaylsis ; neuroimaging ; transposable data

Abstract

Variables in high-dimensional data sets common in neuroimaging and genomics often exhibit complex dependencies. Conventional multivariate analysis methods often ignore these relationships, that can arise, for example, from spatio-temporal processes or network structures. We propose a generalization of the SVD that is appropriate for transposable matrix data, or data in which neither the rows nor columns can be considered independent instances. By finding the best low rank approximation of the data with respect to a transposable quadratic norm, our decomposition, entitled the Generalized least squares Matrix Decomposition (GMD), directly accounts for dependencies in the data. We also regularize the factors, introducing the Generalized Penalized Matrix Factorization (GPMF). We develop fast algorithms using the GMD to preform Generalized PCA (GPCA) and the GPMF to preform sparse GPCA and functional GPCA on massive data sets. Through simulations and a whole-brain functional MRI example 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.