Randomized estimation of functional covariance operator via subsampling

Covariance operators are fundamental concepts and modelling tools for many functional data analysis methods, such as functional principal component analysis. However, the empirical (or estimated) covariance operator becomes too costly to compute when the functional dataset gets big. This paper studies a randomized algorithm for covariance operator estimation. The algorithm works by sampling and rescaling observations from the large functional data collection to form a sketch of much smaller size and performs computation on the sketch to obtain the subsampled empirical covariance operator. The proposed algorithm is theoretically justified via nonasymptotic bounds between the subsampled and the full‐sample empirical covariance operator in terms of the Hilbert‐Schmidt norm and the operator norm. It is shown that the optimal sampling probability that minimizes the expected squared Hilbert‐Schmidt norm of the subsampling error is determined by the norm of each function. Simulated and real data examples are used to illustrate the effectiveness of the proposed algorithm.