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 and Yin, 2013, 2016). However, the DCOV-based SDR method involves optimizing a nonsmooth and nonconvex objective function over the Stiefel manifold. To tackle the numerical challenge, we novelly formulate the original objective function equivalently into a DC (Difference of Convex functions) program and construct an iterative algorithm based on the majorization-minimization (MM) principle. At each step of the MM algorithm, we inexactly solve the quadratic subproblem on 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 the proposed algorithm under some regularity conditions. Simulation studies show our algorithm drastically improves the computation efficiency and is robust across various settings compared with the existing method. Supplemental materials for this article are available.

中文翻译：

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基于距离协方差充分降维和充分变量选择的MM算法

最近提出了使用距离协方差 (DCOV) 的充分降维 (SDR) 作为降维问题的一种方法。与其他 SDR 方法相比，它是无模型的，无需估计链接函数，并且不需要任何特定的预测变量分布（参见 Sheng 和 Yin，2013，2016）。然而，基于 DCOV 的 SDR 方法涉及优化 Stiefel 流形上的非光滑和非凸目标函数。为了应对数值挑战，我们新颖地将原始目标函数等价地公式化为 DC（凸函数差）程序，并基于主最小化（MM）原理构建迭代算法。在 MM 算法的每一步，我们通过对黎曼牛顿方法进行一次迭代，不精确地求解 Stiefel 流形上的二次子问题。该算法还可以使用距离协方差很容易地扩展到足够的变量选择 (SVS)。我们在一定的规律性条件下建立了所提出算法的收敛性。仿真研究表明，与现有方法相比，我们的算法大大提高了计算效率，并且在各种设置下都具有鲁棒性。本文的补充材料可用。