Sparsity oracle inequalities via l_1 regularization in nonparametric models

这一研究工作刻画了一般模型类的稀疏模型恢复的特征，扩展了已有的广义线性模型的结果和分类。这项提案的中心目标是：(A)在高维-低样本量的情况下界定稀疏性和目标回收；(B)表明在最小假设下，具有套索式惩罚的经验风险最小化允许目标回收。研究人员提出了一种新类型的Oracle不等式的使用，以表明惩罚的经验风险最小化因子适应基本统计模型的未知稀疏性。高维数据在许多科学学科中越来越常见，例如生物科学和医学。准确估计和实施用于分析这类数据的复杂统计模型是具有挑战性的。该项目的目的是为分析高维数据环境中计算效率高的程序开发一个统一的理论。这些技术的有效性将通过对基因表达数据和并发的EEG/fMRI数据的应用来证明。新引入的分类程序有一个内置的拒绝选项，允许在难以分类的情况下扣留决定。这极大地提高了误诊后果严重的肿瘤分类的性能。这些程序的软件将在万维网上免费提供。

This research effort characterizes sparse model recovery for general model classes, extending existing results for generalized linear models and in classification. The central goals of this proposal are: (a) to define sparsity and target recovery in high dimensional - low sample size settings; (b) to show that empirical risk minimization with a lasso-type penalty allows for target recovery, under minimal assumptions. The investigators advance the use of a novel type of oracle inequalities to show that the penalized empirical risk minimizers adapt to the unknown sparsity of the underlying statistical model. Special attention is given to random design regression and classification with a reject option.High dimensional data are increasingly common in many scientific disciplines such as biological and medical sciences. Accurate estimation and implementation of complex statistical models used for the analysis of such data are challenging. The aim of this project is to develop a unified theory for the analysis of computationally efficient procedures in high dimensional data settings. The usefulness of these techniques will be demonstrated by applications to gene expression data and concurrent EEG / fMRI data. The newly introduced classification procedures have a built-in reject option that allows for withholding decision in cases that are hard to classify. This greatly improves the performance of tumor classification where the consequences of misdiagnosis are severe. The software for these procedures will be made freely available on the world wide web.