MM algorithms for distance covariance based sufficient dimension reduction and sufficient variable selection

Sufficient dimension reduction (SDR) using distance covariance (DCOV) was recently proposed as an approach to dimension-reduction problems. Compared with other SDR methods, it is model-free without estimating link function and does not require any particular distributions on predictors (see Sheng Yin, 2013, 2016). However, the DCOV-based method involves optimizing a nonsmooth nonconvex objective over Stiefel manifold. To tackle numerical challenge, we novelly formulate original equivalently into DC (Difference of Convex functions) program construct iterative algorithm based majorization-minimization (MM) principle. At each step MM algorithm, inexactly solve quadratic subproblem manifold by taking one iteration Riemannian Newton's method. The can also be readily extended sufficient variable selection (SVS) covariance. We establish convergence property under some regularity conditions. Simulation studies show our drastically improves computation efficiency robust across various settings compared existing Supplemental materials for this article are available.