Summarizing the effect of many covariates through a few linear combinations is an effective way of reducing covariate dimension and is the backbone of (sufficient) dimension reduction. Because the replacement of high-dimensional covariates by low-dimensional linear combinations is performed with a minimum assumption on the specific regression form, it enjoys attractive advantages as well as encounters unique challenges in comparison with the variable selection approach. We review the current literature of dimension reduction with an emphasis on the two most popular models, where the dimension reduction affects the conditional distribution and the conditional mean, respectively. We discuss various estimation and inference procedures in different levels of detail, with the intention of focusing on their underneath idea instead of technicalities. We also discuss some unsolved problems in this area for potential future research.

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降维综述

通过几个线性组合来总结多个协变量的效果，是降低协变量维数的有效方式，也是（充分）降维的支柱。由于用低维线性组合替换高维协变量是在特定回归形式的最小假设下执行的，因此与变量选择方法相比，它具有诱人的优势，但也遇到了独特的挑战。我们回顾了当前关于降维的文献，重点介绍了两个最流行的模型，其中降维分别影响条件分布和条件均值。我们在不同的细节层次上讨论各种估计和推理程序，目的是关注它们的基本思想而不是技术细节。