  Principal component analysis (PCA) is a versatile tool to reduce the
dimensionality which has wide applications in statistics and machine learning
community. It is particularly useful to model data in high-dimensional
scenarios where the number of variables $p$ is comparable to, or much larger
than the sample size $n$. Despite extensive literature on this topic,
researches have focused on modeling static principal eigenvectors or subspaces,
which is unsuitable for stochastic processes that are dynamic in nature. To
characterize the change in the whole course of high-dimensional data
collection, we propose a unified framework to estimate dynamic principal
subspaces spanned by leading eigenvectors of covariance matrices. In the
proposed framework, we formulate an optimization problem by combining the
kernel smoothing and regularization penalty together with the orthogonality
constraint, which can be effectively solved by the proximal gradient method for
manifold optimization. We show that our method is suitable for high-dimensional
data observed under both common and irregular designs. In addition, theoretical
properties of the estimators are investigated under $l_q (0 \leq q \leq 1)$
sparsity. Extensive experiments demonstrate the effectiveness of the proposed
method in both simulated and real data examples.
