Sparse predictors in functional data analysis

Proposed research is motivated from the discrimination task with high dimension, low sample size data. The investigator studies the intrinsic difficulties of the discrimination problem by exploring asymptotic geometric structure of such data. Three main activities are proposed: a) the asymptotic inconsistency of leave-one-out cross-validation. The study is expected to explain why it shall fail when the number of variables greatly exceeds the number of observations; b) the effect of the relationship between the dimensionality and the sample size on the difficulty of discrimination task; c) a discriminant direction vector that only exists for the data with high dimension, low sample size. The data points collapse on this direction vector and also are most separated by group labels. The investigator explores its various theoretical and empirical properties such as its optimality, uniqueness, and asymptotic performances. Even though these topics are loosely related one another in their technical aspects, their goals are essentially the same: exploring the nontraditional and unique challenges in high dimension, low sample size discrimination. While it has been an actively researched area over recent years, however, understanding fundamental challenges of high dimension, low sample size problems is yet satisfactory. This research approaches this problem in a way that may be regarded atypical in a traditional sense, but is more relevant to the problem itself. The applications of proposed research include text document classification such as Spam email filter, medical imaging such as functional magnetic resonance imaging, and bioinformatics such as microarray gene expression and proteomics.