Penalized Principal Component Analysis using Nesterov Smoothing
Rebecca M. Hurwitz Georg Hahn
Principal components computed via PCA (principal component analysis) are
traditionally used to reduce dimensionality in genomic data or to correct for
population stratification. In this paper, we explore the penalized eigenvalue
problem (PEP) which reformulates the computation of the first eigenvector as an
optimization problem and adds an L1 penalty constraint. The contribution of our
article is threefold. First, we extend PEP by applying Nesterov smoothing to
the original LASSO-type L1 penalty. This allows one to compute analytical
gradients which enable faster and more efficient minimization of the objective
function associated with the optimization problem. Second, we demonstrate how
higher order eigenvectors can be calculated with PEP using established results
from singular value decomposition (SVD). Third, using data from the 1000 Genome
Project dataset, we empirically demonstrate that our proposed smoothed PEP
allows one to increase numerical stability and obtain meaningful eigenvectors.
We further investigate the utility of the penalized eigenvector approach over
traditional PCA.