  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.
