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Sufficient dimension reduction (SDR) using distance covariance (DCOV) wasrecently proposed as an approach to dimension-reduction problems. Compared withother SDR methods, it is model-free without estimating link function and doesnot require any particular distributions on predictors (see Sheng and Yin,2013, 2016). However, the DCOV-based SDR method involves optimizing a nonsmoothand nonconvex objective function over the Stiefel manifold. To tackle thenumerical challenge, we novelly formulate the original objective functionequivalently into a DC (Difference of Convex functions) program and constructan iterative algorithm based on the majorization-minimization (MM) principle.At each step of the MM algorithm, we inexactly solve the quadratic subproblemon the Stiefel manifold by taking one iteration of Riemannian Newton's method.The algorithm can also be readily extended to sufficient variable selection(SVS) using distance covariance. We establish the convergence property of theproposed algorithm under some regularity conditions. Simulation studies showour algorithm drastically improves the computation efficiency and is robustacross various settings compared with the existing method. Supplementalmaterials for this article are available.

MM Algorithms for Distance Covariance based Sufficient Dimension Reduction and Sufficient Variable Selection