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基于随机投影的稀疏切片逆回归方法，适用于大$p$小$n$场景。该方法嵌入在广义特征值框架中，最终降维为并行执行低维（广义）特征值分解，从而大幅提升计算效率。在理论上，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，我们的算法包含一种精巧的重新加权机制，可显著增强协变量活动集的可识别性。大量数值研究证明，与竞争方法相比，所提算法具有高度的优越性。