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In this paper, we target the problem of sufficient dimension reduction withsymmetric positive definite matrices valued responses. We propose the intrinsicminimum average variance estimation method and the intrinsic outer productgradient method which fully exploit the geometric structure of the Riemannianmanifold where responses lie. We present the algorithms for our newly developedmethods under the log-Euclidean metric and the log-Cholesky metric. Each of thetwo metrics is linked to an abelian Lie group structure that transforms ourmodel defined on a manifold into a Euclidean one. The proposed methods are thenfurther extended to general Riemannian manifolds. We establish rigourousasymptotic results for the proposed estimators, including the rate ofconvergence and the asymptotic normality. We also develop a cross validationalgorithm for the estimation of the structural dimension with theoreticalguarantee Comprehensive simulation studies and an application to the New Yorktaxi network data are performed to show the superiority of the proposedmethods.

Intrinsic minimum average variance estimation for sufficient dimension reduction with symmetric positive definite matrices and beyond