In this paper, we target the problem of sufficient dimension reduction with symmetric positive definite matrices valued responses. We propose the intrinsic minimum average variance estimation method and the intrinsic outer product gradient method which fully exploit the geometric structure of the Riemannian manifold where responses lie. We present the algorithms for our newly developed methods under the log-Euclidean metric and the log-Cholesky metric. Each of the two metrics is linked to an abelian Lie group structure that transforms our model defined on a manifold into a Euclidean one. The proposed methods are then further extended to general Riemannian manifolds. We establish rigourous asymptotic results for the proposed estimators, including the rate of convergence and the asymptotic normality. We also develop a cross validation algorithm for the estimation of the structural dimension with theoretical guarantee Comprehensive simulation studies and an application to the New York taxi network data are performed to show the superiority of the proposed methods.

翻译：针对响应变量为对称正定矩阵的充分降维问题，本文提出内在最小平均方差估计方法与内在外积梯度法。这两种方法充分利用了响应变量所在黎曼流形的几何结构。我们给出了在新提出的log-欧几里得度量与log-乔莱斯基度量框架下的算法实现：这两种度量均与阿贝尔李群结构相关联，可将定义在流形上的模型转化为欧几里得空间中的问题。进而将所提方法推广至一般黎曼流形。建立了所提估计量的严格渐近理论，包括收敛速率与渐近正态性。我们同时开发了具有理论保证的结构维数估计交叉验证算法。通过综合数值模拟与纽约出租车网络数据的实际应用，验证了所提方法的优越性。