We propose a novel sparse sliced inverse regression method based on random projections in a large $p$ small $n$ setting. Embedded in a generalized eigenvalue framework, the proposed approach finally reduces to parallel execution of low-dimensional (generalized) eigenvalue decompositions, which facilitates high computational efficiency. Theoretically, we prove that this method achieves the minimax optimal rate of convergence under suitable assumptions. Furthermore, our algorithm involves a delicate reweighting scheme, which can significantly enhance the identifiability of the active set of covariates. Extensive numerical studies demonstrate high superiority of the proposed algorithm in comparison to competing methods.

翻译：我们提出了一种新颖的稀疏切片逆回归方法，该方法基于大$p$小$n$场景下的随机投影。嵌入在广义特征值框架中，所提方法最终简化为低维（广义）特征值分解的并行执行，从而实现了高计算效率。理论上，我们证明了该方法在适当假设下达到了极小化最优收敛速率。此外，我们的算法包含一个精细的重新加权方案，可显著增强协变量活跃集的可识别性。大量数值实验表明，与竞争方法相比，所提算法具有高度优越性。