  Functional principal component analysis (FPCA) is a fundamental tool and has
attracted increasing attention in recent decades, while existing methods are
restricted to data with a single or finite number of random functions (much
smaller than the sample size $n$). In this work, we focus on high-dimensional
functional processes where the number of random functions $p$ is comparable to,
or even much larger than $n$. Such data are ubiquitous in various fields such
as neuroimaging analysis, and cannot be properly modeled by existing methods.
We propose a new algorithm, called sparse FPCA, which is able to model
principal eigenfunctions effectively under sensible sparsity regimes. While
sparsity assumptions are standard in multivariate statistics, they have not
been investigated in the complex context where not only is $p$ large, but also
each variable itself is an intrinsically infinite-dimensional process. The
sparsity structure motivates a thresholding rule that is easy to compute
without nonparametric smoothing by exploiting the relationship between
univariate orthonormal basis expansions and multivariate Kahunen-Lo\`eve (K-L)
representations. We investigate the theoretical properties of the resulting
estimators, and illustrate the performance with simulated and real data
examples.
