Spline Estimation of Functional Principal Components via Manifold Conjugate Gradient Algorithm

TL;DR

This paper introduces a manifold conjugate gradient algorithm for functional principal component analysis that leverages geometric structure to enhance estimation accuracy and efficiency, outperforming traditional methods.

Contribution

It proposes a novel conjugate gradient algorithm on the manifold for FPC estimation, improving over EM and REML methods by exploiting geometric structure and allowing flexible penalization.

Findings

Algorithm demonstrates superior accuracy in simulations

Method shows stable and efficient convergence

Application to supernova data validates practical effectiveness

Abstract

Functional principal component analysis has become the most important dimension reduction technique in functional data analysis. Based on B-spline approximation, functional principal components (FPCs) can be efficiently estimated by the expectation-maximization (EM) and the geometric restricted maximum likelihood (REML) algorithms under the strong assumption of Gaussianity on the principal component scores and observational errors. When computing the solution, the EM algorithm does not exploit the underlying geometric manifold structure, while the performance of REML is known to be unstable. In this article, we propose a conjugate gradient algorithm over the product manifold to estimate FPCs. This algorithm exploits the manifold geometry structure of the overall parameter space, thus improving its search efficiency and estimation accuracy. In addition, a distribution-free interpretation of the loss function is provided from the viewpoint of matrix Bregman divergence, which explains why the proposed method works well under general distribution settings. We also show that a roughness penalization can be easily incorporated into our algorithm with a potentially better fit. The appealing numerical performance of the proposed method is demonstrated by simulation studies and the analysis of a Type Ia supernova light curve dataset.