Regularization for High Dimensional Inference and Sparse Recovery

本提案旨在回答在正则化方法的背景下，当简单的最小二乘损失函数不太适合时，识别高维协变量空间的稀疏子集的相关问题。总体目标是了解统计模型的非线性和/或删节结构、正则化方案和问题的内在维度之间的基本相互作用。具体而言，PI旨在(1)识别新的统计问题和正则化方案，这些统计问题和正则化方案具有复杂的非线性和时间-事件结构，具有根据数据特征调整的高维协变量空间；(2)在正则化估计量的行为上发展新的非渐近预言界，其中将利用随机矩阵理论的技术，特别是各种矩阵规范上的高概率界和近似理论，来增强对维数对非渐近性质影响的理解；(3)研究了统计模型可能错误指定的半参数方法风险的新的非渐近界；(4)分析和开发利用问题的审查率、样本量和维度之间的特殊相互作用的模型，重要的是(5)引入最优和有效地解决所研究的大规模问题的新算法。微阵列技术的发展带来了大量的大规模全基因组关联研究，其中同时分析大量snp与发现复杂疾病的遗传鉴定有关。时间对事件分量的存在和重要性要求在NP维度和审查结构的统计方法上取得重大进展。这项研究计划旨在开发创新和有效的统计方法，以处理在遗传、公共卫生和生物信息学领域具有特殊影响的复杂数据，在这些领域，审查和大量基因相互作用使得识别行为不良的基因非常困难。此外，这个跨学科项目的发展将加强新的科学发现，与实践者建立新的合作联系，并将促进当代最先进的机器学习技术应用于半参数模型和审查数据的研究生的教学和培训。为了促进科学的进步，PI将与加州大学圣地亚哥分校医学院数学系、生物统计学和圣地亚哥超级计算机中心进行明确的合作。PI计划通过传播该提案的结果，向数学专业的学生介绍生物学，并在代表性不足的群体和数学女性中推广科学。

This proposal aims at answering pertinent questions to identifying sparse subsets of high dimensional covariate spaces, in the context of regularization methods, when simple least square loss functions are not well suited. The broad goal is to understand the fundamental interactions between nonlinear and/or censored structure of the statistical model, the regularization scheme and the intrinsic dimensionality of the problem. Specifically, the PI aims at (1) Identifying new statistical problems and regularization schemes, with complex nonlinear and time-to-event structure with high dimensional covariate space that are tuned to the characteristics of the data; (2) Developing novel non-asymptotic oracle bounds on the behavior of regularized estimators where techniques of random matrix theory, especially high probability bounds on various matrix norms, and approximation theory, will be utilized to enhance understanding of the effects of dimensionality on the non-asymptotic properties; (3) Investigating new non-asymptotic bounds on risk of semiparametric methods where the statistical model is possibly misspecified; (4) Analyzing and developing models that use special interplay between censoring rate, sample size and dimensionality of the problem and importantly (5) Introducing new algorithms that optimally and efficiently solve the investigated large scale problems.Explosion of microarray technologies has lead to vast number of large-scale genome-wide association studies where simultaneous analysis of a large number of SNPs is pertinent to discovering genetic identification of complex diseases. Presence and importance of time to event component calls for significant advances in statistical methodology for both NP dimensionality and censored structure. This research proposal aims at developing innovative and effective statistical methods for such complex data with special impact in genetic, public health and bioinformatic sciences, where censoring and vast number of gene interactions make identification of misbehaving genes very difficult. Moreover, the developments of this inter-disciplinary project will enhance new scientific discoveries, make new collaborative connections with practitioners and will promote teaching and training of graduate students on the contemporary state-of-the-art machine learning techniques applied to semiparametric models and censored data. To promote the progress of science, PI will make explicit collaborations of department of Mathematics, Biostatistics division of the Medical School at UCSD and Supercomputer Center in San Diego. Through dissemination of the results of this proposal, PI plans to expose biology to mathematics majors and promote science among underrepresented groups and women in mathematics.