主 题：A unified generalization of inverse regression via adaptive column selection

主讲人：浙江大学骆威博士

主持人：统计学院周岭副教授

时间：2022年12月20日（周二）上午10:30-11:30

直播平台及会议ID：腾讯会议，ID: 823-588-524

主办单位：统计研究中心和统计学院 科研处

主讲人简介：

骆威,2014年毕业于美国宾夕法尼亚州立大学，之后任职于美国Baruch College,于2018年加入浙江大学。骆威的研究方向包括充分降维和因果推断，在Annals of Statistics, Biometrika, JRSSB等统计国际学术期刊上发表了多篇论文，目前主持国家优秀青年科学基金项目。

内容提要：

Higher-order inverse regression methods are commonly known as more powerful sufficient dimension reduction (SDR) methods than the popularly used sliced inverse regression (SIR) in the population level. However, due to the convention of essentially conducting singular value decomposition on the ambient candidate matrices, these methods suffer from the excessive number of parameters in the sample level and have not been systematically generalized under the high-dimensional settings like SIR. In this paper, we break the convention of using the ambient candidate matrices in these methods, and instead apply a novel column-selection strategy on their candidate matrices that substantially lowers down the working number of parameters to being comparable with SIR. Then, for the first time of the literature, we generalize the higher-order inverse regression methods, as well as their ensembles, towards sparsity under the high-dimensional settings in a uniform manner. The dimension of the predictor is allowed to diverge with the sample size in nearly an exponential order, and no additional restrictions are imposed on the data other than those commonly seen in the high-dimensional literature. For completeness of theory, we also study the column-selection strategy towards the estimation efficiency under the conventional low-dimensional settings. These results are illustrated by simulation studies and a real data application at the end.

在总体水平上，高阶逆回归方法通常被称为比普遍使用的切片逆回归(SIR)更强大的充分降维方法。但是，由于这些方法本质上是对环境候选矩阵进行奇异值分解的惯例，导致这些方法在样本水平上参数数量过多，在SIR等高维设置下没有得到系统的推广。在本文中，我们打破了这些方法中使用环境候选矩阵的惯例，并在候选矩阵上应用一种新的列选择策略，大大降低了参数的工作数量，使其与SIR具有可比性。然后，我们首次在文献中推广了高阶逆回归方法及其集合，以统一的方式在高维设置下实现稀疏性。预测器的维数允许随样本量以接近指数级的顺序发散，并且除了在高维文献中常见的数据外，没有对数据施加额外的限制。为了理论的完备性，我们还研究了在常规低维设置下对估计效率的列选择策略。最后通过仿真研究和实际数据应用对这些结果进行了说明。