Regularized Dimension Reduction for High Dimensional Data

随着近年来生物技术的发展，如全基因组微阵列和高通量测序的使用，基于回归的高维数据建模在生物科学中从未如此重要。研究者的目标是发展一个正则化的降维方法来处理非常高维的线性回归问题。该研究的主旨是基于一个行之有效的降维技术，称为偏最小二乘（PLS）回归，已大量使用在几个科学研究领域的不适定问题通常出现。 本文的工作是：1）从理论上研究了偏最小二乘方法在高维回归中的适用性，在高维回归中，预测变量的数量远远超过可用的样本量; 2）提出了一种正则化方案，该方案在降维的同时促进变量的选择; 3）开发了一个有效的算法来实现所提出的框架。还考虑了相关分类和删失数据设置的扩展。建议的工作，完成和传播后，将提供一个强大的同时降维和变量选择框架相关的所有领域的科学研究，关注高维不适定回归问题。这将使科学家能够分析高维数据，有效地降维和提高可解释性。PI积极参与与生物学家，生物化学家，遗传学家和医生的合作。因此，这项建议所产生的研究将具有强烈的跨学科色彩，并将得到实施、测试和调整，以解决许多感兴趣的真实的科学问题。PI将把拟议的研究应用于研究基因表达、转录调控和DNA结合蛋白结合特性的变化中出现的问题，其中相关变量的选择与具有良好的预测能力一样重要。该项目将通过与研究生和本科生密切合作，将研究和教育结合起来。

With the recent advancements in biotechnology such as the use of genomewide microarrays and high throughput sequencing, regression-based modeling of high dimensional data in biological sciences has never been more important. The investigator aims to develop a regularized dimension reduction method for very high dimensional linear regression problems. The main thrust of the research is based on a well-established dimension reduction technique named Partial Least Squares (PLS) regression which has been heavily used in several scientific research areas where ill-posed problems commonly arise. The proposed work 1) theoretically investigates the suitability of PLS for very high dimensional regression settings where the number of predictors highly exceeds the available sample size; 2) proposes a regularization scheme that promotes variable selection in addition to dimension reduction; constructs rigorous mathematical formulations of the regularization scheme and characterizes their analytical solutions; 3) develops an efficient algorithm implementing the proposed framework. Extensions to interrelated classification and censored data settings are also considered. The proposed work, when completed and disseminated, will provide a powerful simultaneous dimension reduction and variable selection framework relevant for all fields of scientific research that concern high dimensional ill-posed regression problems. This will allow scientists to analyze high-dimensional data with efficient dimension reduction and increased interpretability. The PI is actively involved in collaborations with biologists, biochemists, geneticists, and medical doctors. The research emanating from this proposal will therefore have strong interdisciplinary flavor and will be implemented, tested and tuned to address many real scientific questions of interest. The PI will apply the proposed research to problems arising in studying the variation of gene expression, transcription regulation, and binding properties of DNA binding proteins, where the selection of relevant variables is as important as having excellent predictive power. The project will integrate research and education by working closely with both graduate and undergraduate students.