  Principal component regression (PCR) is a two-stage procedure: the first
stage performs principal component analysis (PCA) and the second stage
constructs a regression model whose explanatory variables are replaced by
principal components obtained by the first stage. Since PCA is performed by
using only explanatory variables, the principal components have no information
about the response variable. To address the problem, we propose a one-stage
procedure for PCR in terms of singular value decomposition approach. Our
approach is based upon two loss functions, a regression loss and a PCA loss,
with sparse regularization. The proposed method enables us to obtain principal
component loadings that possess information about both explanatory variables
and a response variable. An estimation algorithm is developed by using
alternating direction method of multipliers. We conduct numerical studies to
show the effectiveness of the proposed method.
