Sparse and Functional Principal Components Analysis

Abstract: 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.