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切片逆回归的结构维度问题AI 解读

In this work, we address the longstanding puzzle that Sliced Inverse Regression (SIR) often performs poorly for sufficient dimension reduction when the structural dimension d (the dimension of the central space) exceeds 4. We first show that in the multiple index model Y=f(PX)+ϵ where X is a p-standard normal vector, ϵ is an independent noise, and P is a projection operator from Rp to Rd, if the link function f follows the law of a Gaussian process. Then with high…

高维核插值的相图AI 解读

Summary The generalization ability of kernel interpolation in large dimensions, ie, $ n\asymp d^{\gamma} $ for some $ \gamma \gt 0 $, could be one of the most interesting problems in the recent renaissance of kernel regression, since it may help us understand the so-called benign overfitting phenomenon reported in the neural networks literature. Focusing on the inner product kernel on the unit sphere, we fully characterize the exact order of both the variance and the bias of…

核插值的泛化表现不佳AI 解读

Summary One of the most interesting problems in the recent renaissance of the studies in kernel regression might be whether kernel interpolation can generalize well, since it may help us understand the ‘benign overfitting phenomenon’ reported in the literature on deep networks. In this paper, under mild conditions, we show that, for any ε>0, the generalization error of kernel interpolation is lower bounded by Ω(n−ε). In other words, the kernel interpolation generalizes…

The central subspace of a pair of random variables $(y,\boldsymbol{x})\in \mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y\perp\!\!\!\!\!\perp \boldsymbol{x}|P_{\mathcal{S}}\boldsymbol{x}$. In this paper, we consider the minimax rate of estimating the central space under the multiple index model $y=f(\boldsymbol{\beta }_{1}^{\tau }\boldsymbol{x},\boldsymbol{\beta }_{2}^{\tau }\boldsymbol{x},\ldots,\boldsymbol{\beta }_{d}^{\tau }\boldsymbol{x},\epsilon )$ w…

同伴效应的随机化推断AI 解读

Xinran Lia, Peng Dingb*, Qian Linc, Dawei Yangd & Jun S. Liuaa Department of Statistics, Harvard University, Cambridge, MA; b Department of Statistics, University of California, Berkeley, CA; c Center for Statistical Science, Department of Industrial Engineering, Tsinghua University, Beijing, P. R. China; d Bureau of Personnel of Chinese Academy of Sciences & School of Education of Peking University, Beijing, P. R. ChinaSupplementary materials for this article are available…

基于 Lasso 的稀疏切片逆回归AI 解读

For multiple index models, it has recently been shown that the sliced inverse regression (SIR) is consistent for estimating the sufficient dimension reduction (SDR) space if and only if $ρ = \mathrm{lim} \frac{p}{n} = 0$, where p is the dimension and n is the sample size. Thus, when p is of the same or a higher order of n, additional assumptions such as sparsity must be imposed in order to ensure consistency for SIR. By constructing artificial response variables made up from…

高维切片逆回归的一致性与稀疏性AI 解读

We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by Li [J. Amer. Statist. Assoc. 86 (1991) 316–342]. Under mild conditions, the asymptotic ratio $\rho=\lim p/n$ is the phase transition parameter and the SIR estimator is consistent if and only if $\rho=0$. When dimension $p$ is greater than $n$, we propose a diagonal thresholding screening SIR (DT-SIR) algorithm.…