  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.
