分组和分层变量选择的复合绝对罚函数族
The composite absolute penalties family for grouped and hierarchical variable selection
摘要：
从高维数据中提取有用的信息是当今统计研究和实践的重要焦点。惩罚损失函数最小化已被证明是有效的，这一任务的理论和经验。由于具有正则化和稀疏性的优点，$L_1$ -惩罚平方误差最小化方法Lasso在回归模型和其他模型中很受欢迎。本文将包括$L_1$在内的不同范数组合起来，形成一种智能惩罚，以便在回归或分类模型的拟合中加入边信息，从而获得合理的估计。具体来说，我们引入了复合绝对惩罚（CAP）家庭，它允许给定的分组和层次关系的预测要表达的。CAP惩罚是通过定义群体并在跨群体和群体内水平上结合规范惩罚的属性来构建的。分组选择用于不重叠的组。通过定义具有特定重叠模式的组来实现分层变量选择。我们建议使用BLASSO和交叉验证来计算CAP估计。对于只涉及$L_1$和$L_{\infty}$范数的CAP估计的子类，我们引入iCAP算法来跟踪分组选择问题的整个正则化路径。在这个子族中，自由度（df）的无偏估计推导出来，使得正则化参数的选择没有交叉验证。CAP在一系列模拟实验中，包括$p\gg n$和可能错误指定分组的情况下，显示出对LASSO的预测性能的改善。当模型的复杂性被正确计算时，iCAP在实验中被认为是吝啬的。
英文摘要：
Extracting useful information from high-dimensional data is an important focus of today's statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the $L_1$-penalized squared error minimization method Lasso has been popular in regression models and beyond. In this paper, we combine different norms including $L_1$ to form an intelligent penalty in order to add side information to the fitting of a regression or classification model to obtain reasonable estimates. Specifically, we introduce the Composite Absolute Penalties (CAP) family, which allows given grouping and hierarchical relationships between the predictors to be expressed. CAP penalties are built by defining groups and combining the properties of norm penalties at the across-group and within-group levels. Grouped selection occurs for nonoverlapping groups. Hierarchical variable selection is reached by defining groups with particular overlapping patterns. We propose using the BLASSO and cross-validation to compute CAP estimates in general. For a subfamily of CAP estimates involving only the $L_1$ and $L_{\infty}$ norms, we introduce the iCAP algorithm to trace the entire regularization path for the grouped selection problem. Within this subfamily, unbiased estimates of the degrees of freedom (df) are derived so that the regularization parameter is selected without cross-validation. CAP is shown to improve on the predictive performance of the LASSO in a series of simulated experiments, including cases with $p\gg n$ and possibly mis-specified groupings. When the complexity of a model is properly calculated, iCAP is seen to be parsimonious in the experiments.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
