Sampled forms of functional PCA in reproducing kernel Hilbert spaces

TL;DR

This paper investigates the effects of limited functional sampling on PCA in reproducing kernel Hilbert spaces, providing convergence rates and optimal bounds for a regularized PCA estimator.

Contribution

It introduces a model for sampling in RKHS, analyzes the convergence of a regularized PCA estimator, and establishes minimax optimal rates in a nonasymptotic setting.

Findings

Convergence rates depend on the number of samples n and m.

Regularized PCA is computationally attractive and effective.

Derived minimax lower bounds match the estimator's performance.

Abstract

We consider the sampling problem for functional PCA (fPCA), where the simplest example is the case of taking time samples of the underlying functional components. More generally, we model the sampling operation as a continuous linear map from $\mathcal{H}$ to $\mathbb{R}^m$, where the functional components to lie in some Hilbert subspace $\mathcal{H}$ of $L^2$, such as a reproducing kernel Hilbert space of smooth functions. This model includes time and frequency sampling as special cases. In contrast to classical approach in fPCA in which access to entire functions is assumed, having a limited number m of functional samples places limitations on the performance of statistical procedures. We study these effects by analyzing the rate of convergence of an M-estimator for the subspace spanned by the leading components in a multi-spiked covariance model. The estimator takes the form of regularized PCA, and hence is computationally attractive. We analyze the behavior of this estimator within a nonasymptotic framework, and provide bounds that hold with high probability as a function of the number of statistical samples n and the number of functional samples m. We also derive lower bounds showing that the rates obtained are minimax optimal.