Forward Regression via Gram-Schmidt Orthogonalization for Ultra-High Dimensional Linear Models
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超高维线性模型中基于 Gram–Schmidt 正交化的前向回归
- 研究问题
- 超高维且预测变量高度共线时，如何降低前向回归反复投影的计算成本与数值不稳定，并选择合适的模型规模？
- 方法与思路
- 作者用递归 Gram–Schmidt 更新实现正交化前向回归，按新增独立贡献排序变量，并利用前向路径统计量选择模型规模。
- 材料陈述的贡献
- 摘要称推导最优收敛率，并在适当条件下获得可靠筛选和变量选择一致性；模拟与实例展示计算、筛选及预测表现。
- 阅读关注点
- 应核对最优收敛率对应的损失与模型类别、共线性允许程度、筛选和选择条件，以及路径停止规则的实现。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未经核验；摘要未给出具体条件和比较数值，不能据此认定对任意共线设计有效。
原文摘要
Forward regression is a classical and effective tool for variable screening in ultra-high dimensional linear models, but its standard projection-based implementation can be computationally costly and numerically unstable when predictors are strongly collinear. Motivated by this limitation, we propose an orthogonalized forward regression procedure, implemented recursively through Gram-Schmidt updates, that ranks predictors according to their unique contributions after removing the effects of variables already selected. This approach preserves the interpretability of forward regression while substantially reducing the cost of repeated projections. We further develop a path-based model size selection rule using statistics computed directly from the forward sequence, thereby avoiding cross-validation and extensive tuning. The resulting method is particularly well suited to settings in which the number of predictors far exceeds the sample size and strong collinearity renders the conventional forward fitting ineffective. Theoretically, we derive the optimal convergence rate for the proposed Gram-Schmidt forward regression, thereby extending existing results for projection-based forward regression, and further show that it enjoys sure screening property and variable selection consistency under suitable conditions. Simulation studies and empirical examples demonstrate that it provides a favorable balance among computational efficiency, numerical stability, screening accuracy, and predictive performance, especially in highly correlated ultra-high dimensional settings.