Sparse kernel sufficient dimension reduction.

The sufficient dimension reduction (SDR) with sparsity has received much attention for analysing high-dimensional data. We study a nonparametric sparse kernel sufficient dimension reduction (KSDR) based on the reproducing kernel Hilbert space, which extends the methodology of the sparse SDR based on inverse moment methods. We establish the statistical consistency and efficient estimation of the sparse KSDR under the high-dimensional setting where the dimension diverges as the sample size increases. Computationally, we introduce a new nonconvex alternating directional method of multipliers (ADMM) to solve the challenging sparse SDR and propose the nonconvex linearised ADMM to solve the more challenging sparse KSDR. We study the computational guarantees of the proposed ADMMs and show an explicit iteration complexity bound to reach the stationary solution. We demonstrate the finite-sample properties in simulation studies and a real application.