Abstract: Existing functional principal component analysis (FPCA) 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 modeled properly by existing methods. We propose a new algorithm, called sparse FPCA, that models principal eigenfunctions effectively under sensible sparsity regimes. The sparsity structure motivates a thresholding rule that is easy to compute by exploiting the relationship between univariate orthonormal basis expansions and the multivariate Karhunen-Loève representation. We investigate the theoretical properties of the resulting estimators, and illustrate the performance using simulated and real-data examples.

Key words and phrases: Basis expansion, multivariate Karhunen-Loève expansion, sparsity regime.