Kernel Additive Principal Components

TL;DR

This paper introduces a kernel-based regularized method for estimating additive principal components, enabling flexible modeling of nonlinear data constraints while ensuring computational feasibility and theoretical consistency.

Contribution

It develops a novel kernel-based approach for APC estimation that allows infinite-dimensional function spaces and proves its consistency.

Findings

Kernel methods enable flexible nonlinear APC modeling.

The proposed approach is computationally feasible.

Consistency of the estimated APCs is established.

Abstract

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.