Testing Model Utility for Single Index Models Under High Dimension
Testing Model Utility for Single Index Models Under High Dimension
复制标题
高维下单指标模型的模型实用性测试
DOI：
10.1007/978-3-030-69009-0_4
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Liu, Jun S
中科院分区：
文献类型：
--
作者：
Lin, Qian;Zhao, Zhigen;Liu, Jun S
For the single index modely=f(βτx,𝜖) with Gaussian design, wherefis unknown andβis a sparsep-dimensional unit vector with at mostsnonzero entries, we are interested in testing the null hypothesis thatβ, when viewed as a whole vector, is zero against the alternative that some entries ofβis nonzero, withni.i.d observations. Assuming that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$var(\mathbb {E}[\boldsymbol {x} \mid y])$$ \end{document} is non-vanishing, we define the generalized signal-to-noise ratio (gSNR)λof the model as the unique non-zero eigenvalue of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$var(\mathbb {E}[\boldsymbol {x} \mid y])$$ \end{document}. We show that ifis of a smaller order ofn, denoted as, one can detect the existence of signals if and only if gSNR. Furthermore, if the noise is additive (i.e.,y=f(βτx) +𝜖), one can detect the existence of the signal if and only if gSNR. It is rather surprising that the detection boundary for the single index model with additive noise matches that for linear regression models. These results pave the road for thorough theoretical analysis of single/multiple index models in high dimensions.
For the single index modely=f(βτx,𝜖) with Gaussian design, wherefis unknown andβis a sparsep-dimensional unit vector with at mostsnonzero entries, we are interested in testing the null hypothesis thatβ, when viewed as a whole vector, is zero against the alternative that some entries ofβis nonzero, withni.i.d observations. Assuming that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$var(\mathbb {E}[\boldsymbol {x} \mid y])$$ \end{document} is non-vanishing, we define the generalized signal-to-noise ratio (gSNR)λof the model as the unique non-zero eigenvalue of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$var(\mathbb {E}[\boldsymbol {x} \mid y])$$ \end{document}. We show that ifis of a smaller order ofn, denoted as, one can detect the existence of signals if and only if gSNR. Furthermore, if the noise is additive (i.e.,y=f(βτx) +𝜖), one can detect the existence of the signal if and only if gSNR. It is rather surprising that the detection boundary for the single index model with additive noise matches that for linear regression models. These results pave the road for thorough theoretical analysis of single/multiple index models in high dimensions.
影响因子：
1.1
作者：
Ingster, Yuri I.;Tsybakov, Alexandre B.;Verzelen, Nicolas
通讯作者：
Verzelen, Nicolas
DOI：
10.1214/19-aos1813
发表时间：
2017-01
期刊：
The Annals of Statistics
影响因子：
--
作者：
Q. Lin;Xinran Li;Dongming Huang;Jun S. Liu
通讯作者：
Q. Lin;Xinran Li;Dongming Huang;Jun S. Liu
分析 fonctionnelle / 泛函分析 具有独立列的随机矩阵的最小奇异值
DOI：
--
发表时间：
2009
期刊：
影响因子：
--
作者：
Rados law;O. Guédon;A. Litvak;A. Pajor;N. Tomczak
通讯作者：
N. Tomczak