  Additive principal components (APCs for short) are a nonlinear generalization
of linear principal components. We focus on smallest APCs to describe additive
nonlinear constraints that are approximately satisfied by the data. Thus APCs
fit data with implicit equations that treat the variables symmetrically, as
opposed to regression analyses which fit data with explicit equations that
treat the data asymmetrically by singling out a response variable. We propose a
regularized data-analytic procedure for APC estimation using kernel methods. In
contrast to existing approaches to APCs that are based on regularization
through subspace restriction, kernel methods achieve regularization through
shrinkage and therefore grant distinctive flexibility in APC estimation by
allowing the use of infinite-dimensional functions spaces for searching APC
transformation while retaining computational feasibility. To connect population
APCs and kernelized finite-sample APCs, we study kernelized population APCs and
their associated eigenproblems, which eventually lead to the establishment of
consistency of the estimated APCs. Lastly, we discuss an iterative algorithm
for computing kernelized finite-sample APCs.
