UPS delivers optimal phase diagram in high-dimensional variable selection
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UPS 实现高维变量选择的最优相图
- 研究问题
- 在变量数超过样本量、信号稀疏且设计相关矩阵未知的高维线性模型中，何时能够恢复全部或大部分非零系数？
- 方法与思路
- 作者提出 UPS，先用单变量阈值筛选，再以惩罚最大似然清理；利用筛选后的可分离性，将大问题拆成多个小回归。
- 材料陈述的贡献
- 摘要称 UPS 在许多情形下达到 Hamming 损失的最优收敛率，并呈现完整恢复、部分恢复和无法成功选择的三阶段相图。
- 阅读关注点
- 应核对设计相关矩阵限制、信号强度与稀疏度的渐近标度、筛选和分解性质，以及最优率和比较方法失效区域的条件。
- 阅读边界
- 版本未注明的索引摘要未给出完整参数区域和正则条件；不能把若干情形下的最优性扩展到任意设计或信号结构。
原文摘要
Consider a linear model Y = Xβ + z, z ∼ N(0, In). Here, X = Xn,p, where both p and n are large, but p > n. We model the rows of X as i.i.d. samples from N(0, 1/n Ω), where Ω is a p × p correlation matrix, which is unknown to us but is presumably sparse. The vector β is also unknown but has relatively few nonzero coordinates, and we are interested in identifying these nonzeros. We propose the Univariate Penalization Screeing (UPS) for variable selection. This is a screen and clean method where we screen with univariate thresholding and clean with penalized MLE. It has two important properties: sure screening and separable after screening. These properties enable us to reduce the original regression problem to many small-size regression problems that can be fitted separately. The UPS is effective both in theory and in computation. We measure the performance of a procedure by the Hamming distance, and use an asymptotic framework where p → ∞ and other quantities (e.g., n, sparsity level and strength of signals) are linked to p by fixed parameters. We find that in many cases, the UPS achieves the optimal rate of convergence. Also, for many different Ω, there is a common three-phase diagram in the two-dimensional phase space quantifying the signal sparsity and signal strength. In the first phase, it is possible to recover all signals. In the second phase, it is possible to recover most of the signals, but not all of them. In the third phase, successful variable selection is impossible. UPS partitions the phase space in the same way that the optimal procedures do, and recovers most of the signals as long as successful variable selection is possible. The lasso and the subset selection are well-known approaches to variable selection. However, somewhat surprisingly, there are regions in the phase space where neither of them is rate optimal, even in very simple settings, such as Ω is tridiagonal, and when the tuning parameter is ideally set.