Abstract: This paper considers smooth principle component analysis for high di- mensional data with very large dimensional observations p and moderate number of individuals N. Our setting is similar to traditional PCA, but we assume the factors are smooth and design a new approach to estimate them. By connecting with Singular Value Decomposition subjected to penalized smoothing, our algorithm is linear in the dimensionality of the data, and it also favors block calculations and sequential access to memory. Di erent from most existing methods, we avoid extracting eignefunctions via smoothing a huge dimensional covariance operator. Under regularity assumptions, the results indicate that we may enjoy faster convergence rate by employing smoothness assumption. We also extend our meth- ods when each subject is given multiple tasks by adopting the two way ANOVA approach to further demonstrate the advantages of our approach.

Key Words: Principal Component Analysis; Penalized Smoothing; Asymp- totics; Multilevel; fMRI;