Principal component analysis and clustering on manifolds

Big data, high dimensional data, sparse data, large scale data, and imaging data are all becoming new frontiers of statistics. Changing technologies have created this flood and have led to a real hunger for new modeling strategies and data analysis by scientists. In many cases data are not Euclidean; for example, in molecular biology, the data sit on manifolds. Even in a simple non-Euclidean manifold (circle), to summarize angles by the arithmetic average cannot make sense and so more care is needed. Thus non-Euclidean settings throw up many major challenges, both mathematical and statistical. This paper will focus on the PCA and clustering methods for some manifolds. Of course, the PCA and clustering methods in multivariate analysis are one of the core topics.