A Generalized Least-Square Matrix Decomposition

Abstract

Variables in many big-data settings are structured, arising, for example, from measurements on a regular grid as in imaging and time series or from spatial-temporal measurements as in climate studies. Classical multivariate techniques ignore these structural relationships often resulting in poor performance. We propose a generalization of principal components analysis (PCA) that is appropriate for massive datasets with structured variables or known two-way dependencies. By finding the best low-rank approximation of the data with respect to a transposable quadratic norm, our decomposition, entitled the generalized least-square matrix decomposition (GMD), directly accounts for structural relationships. As many variables in high-dimensional settings are often irrelevant, we also regularize our matrix decomposition by adding two-way penalties to encourage sparsity or smoothness. We develop fast computational algorithms using our methods to perform generalized PCA (GPCA), sparse GPCA, and functional GPCA on massive datasets. Through simulations and a whole brain functional MRI example, we demonstrate the utility of our methodology for dimension reduction, signal recovery, and feature selection with high-dimensional structured data. Supplementary materials for this article are available online.

KEY WORDS:

• Functional PCA
• Neuroimaging
• Principal components analysis
• Singular value decomposition
• Sparse PCA
• Structured data

Additional information

Funding

The authors thank the editor, associate editor, and two anonymous reviewers for several helpful suggestions. The authors also thank Susan Holmes for bringing relevant references to our attention, and Frederick Campbell for work on software. G. I. Allen is partially supported by NSF DMS-1209017, J. Taylor is partially supported by NSF DMS-0906801, and L. Grosenick is supported by NSF IGERT Award #0801700.