We propose a multistep screening procedure (MuSP) for the recovery of sparse linear models in high‐dimensional data. This method is based on a repeated small penalty strategy that quickly converges to an estimate within a few iterations. Specifically, in each iteration, an adaptive lasso regression with a small penalty is fit within the reduced feature space obtained from the previous step, rendering its computational complexity roughly comparable with the Lasso. MuSP is shown to select the true model under complex correlation structures among the predictors and response, even when the irrepresentable condition fails. Further, under suitable regularity conditions, MuSP achieves the optimal minimax rate ${(q\log n/n)}^{1/2}$ for the upper bound of l₂‐norm error. Numerical comparisons show that the method works effectively both in model selection and estimation, and the MuSP fitted model is stable over a range of small tuning parameter values, eliminating the need to choose the tuning parameter by cross‐validation. We also apply MuSP to financial data and show that MuSP is successful in asset allocation selection.

中文翻译：

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MuSP：用于稀疏恢复的多步骤筛选程序

我们提出了一种多步骤筛选程序（MuSP），用于恢复高维数据中的稀疏线性模型。该方法基于重复的小罚分策略，该策略在几次迭代中快速收敛到估计值。具体而言，在每次迭代中，在前一步骤获得的缩减特征空间内都以较小的代价拟合了自适应套索回归，从而使其计算复杂度与套索大致相当。MuSP被证明可以在预测变量和响应之间的复杂相关结构下选择真实的模型，即使无法表述的条件失败了也是如此。此外，在适当的规则性条件下，MuSP可以达到最佳的最小最大速率${（q\mathrm{日志}ñ/ñ）}^{\mathrm{1个}/\mathrm{2个}}$对于l ₂范数误差的上限。数值比较表明，该方法在模型选择和估计中均有效，并且MuSP拟合模型在较小的调整参数值范围内是稳定的，从而无需通过交叉验证来选择调整参数。我们还将MuSP应用于财务数据，并显示MuSP在资产分配选择方面是成功的。