Sure Independence Screening for Ultra-High Dimensional Feature Space
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超高维特征空间的确定独立筛选
- 研究问题
- 当特征维度可指数增长时，如何先将候选变量缩减到样本量以下，同时保留重要变量，以降低后续变量选择的成本？
- 方法与思路
- 作者提出基于相关性学习的 SIS 筛选，分析确定筛选性质，并提出迭代版本 ISIS，再将筛选结果交由惩罚变量选择方法处理。
- 材料陈述的贡献
- 摘要称，SIS 在一定渐近框架下可处理指数增长维度并保留重要变量，ISIS 用于改善有限样本表现，筛选可提高后续选择的速度与准确性。
- 阅读关注点
- 应核对重要变量保留所需的信号和相关性条件、筛选维度的选择、指数增长的限制，以及 ISIS 与后续选择方法的配合方式。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未核实；摘要未列出完整筛选条件和数值结果，保留重要变量不等于精确恢复全部模型。
原文摘要
Variable selection plays an important role in high dimensional statistical modeling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality $p$, estimation accuracy and computational cost are two top concerns. In a recent paper, Candes and Tao (2007) propose the Dantzig selector using $L_1$ regularization and show that it achieves the ideal risk up to a logarithmic factor $\log p$. Their innovative procedure and remarkable result are challenged when the dimensionality is ultra high as the factor $\log p$ can be large and their uniform uncertainty principle can fail. Motivated by these concerns, we introduce the concept of sure screening and propose a sure screening method based on a correlation learning, called the Sure Independence Screening (SIS), to reduce dimensionality from high to a moderate scale that is below sample size. In a fairly general asymptotic framework, the correlation learning is shown to have the sure screening property for even exponentially growing dimensionality. As a methodological extension, an iterative SIS (ISIS) is also proposed to enhance its finite sample performance. With dimension reduced accurately from high to below sample size, variable selection can be improved on both speed and accuracy, and can then be accomplished by a well-developed method such as the SCAD, Dantzig selector, Lasso, or adaptive Lasso. The connections of these penalized least-squares methods are also elucidated.