Sparse Functional Principal Component Analysis in High Dimensions

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 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 $p$ comparable to, even much larger $n$. Such ubiquitous various fields such as neuroimaging analysis, cannot be properly modeled by methods. We propose new algorithm, called sparse FPCA, which able model eigenfunctions effectively under sensible sparsity regimes. While assumptions standard multivariate statistics, they have not been investigated complex context only large, but also each variable itself an intrinsically infinite-dimensional process. The structure motivates thresholding rule that easy compute without nonparametric smoothing exploiting relationship between univariate orthonormal basis expansions Kahunen-Lo\`eve (K-L) representations. investigate theoretical properties resulting estimators, illustrate performance simulated real examples.