  Variables in many massive high-dimensional data sets 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 the singular value
decomposition (SVD) and principal components analysis (PCA) that is appropriate
for massive data sets 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
squares Matrix Decomposition (GMD), directly accounts for structural
relationships. As many variables in high-dimensional settings are often
irrelevant or noisy, 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 data sets. 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.
