  Regularized variants of Principal Components Analysis, especially Sparse PCA
and Functional PCA, are among the most useful tools for the analysis of complex
high-dimensional data. Many examples of massive data, have both sparse and
functional (smooth) aspects and may benefit from a regularization scheme that
can capture both forms of structure. For example, in neuro-imaging data, the
brain's response to a stimulus may be restricted to a discrete region of
activation (spatial sparsity), while exhibiting a smooth response within that
region. We propose a unified approach to regularized PCA which can induce both
sparsity and smoothness in both the row and column principal components. Our
framework generalizes much of the previous literature, with sparse, functional,
two-way sparse, and two-way functional PCA all being special cases of our
approach. Our method permits flexible combinations of sparsity and smoothness
that lead to improvements in feature selection and signal recovery, as well as
more interpretable PCA factors. We demonstrate the efficacy of our method on
simulated data and a neuroimaging example on EEG data.
