Regularized Dimension Reduction for High Dimensional Data

随着最近生物技术的进步，如全基因组微阵列和高通量测序的使用，生物科学中基于回归的高维数据建模从未如此重要。研究者的目标是发展一种正则化的降维方法，用于非常高维的线性回归问题。该研究的主要推力是基于一种完善的降维技术，即偏最小二乘（PLS）回归，该技术已被广泛应用于一些常出现病态问题的科学研究领域。提出的工作1)从理论上研究PLS对非常高维回归设置的适用性，其中预测因子的数量高度超过可用的样本量；2)提出了一种在降维的同时促进变量选择的正则化方案；构造正则化方案的严格数学公式，并表征其解析解；3)开发了一个有效的算法来实现所提出的框架。扩展到相关的分类和审查数据设置也被考虑。拟议的工作一旦完成和传播，将为涉及高维病态回归问题的所有科学研究领域提供一个强有力的同时降维和变量选择框架。这将使科学家能够分析高维数据，有效地减少维数，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，从该提案中产生的研究将具有强烈的跨学科风味，并将被实施、测试和调整，以解决许多真正感兴趣的科学问题。PI将把所提出的研究应用于研究基因表达变化、转录调节、DNA结合蛋白结合特性等方面出现的问题，在这些问题上，相关变量的选择与具有出色的预测能力同样重要。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.