Sparse predictors in functional data analysis

本研究的动机来自于高维度、低样本量数据的辨别任务。研究者通过探索这些数据的渐近几何结构来研究歧视问题的内在困难。提出了三个主要活动：a）留一交叉验证的渐近不一致性。该研究旨在解释为什么当变量的数量大大超过观察的数量时，它会失败; B）维度和样本大小之间的关系对区分任务难度的影响; c）仅存在于高维，低样本大小的数据的判别方向向量。 数据点在此方向向量上折叠，并且大多数由组标签分隔。研究者探索了它的各种理论和经验性质，如它的最优性，唯一性和渐近性能，即使这些主题在技术方面彼此松散相关，他们的目标基本上是相同的：探索高维，低样本量歧视的非传统和独特的挑战。虽然近年来它一直是一个积极的研究领域，但是，了解高维，低样本量问题的基本挑战仍然令人满意。本研究以一种在传统意义上可能被视为非典型的方式来处理这个问题，但与问题本身更相关。建议的研究的应用包括文本文档分类，如垃圾邮件过滤器，医学成像，如功能磁共振成像，和生物信息学，如微阵列基因表达和蛋白质组学。

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.