  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.
