Abstract: In this paper, we estimate the central mean subspace in a dimension reduction problem where the response is a symmetric positive-definite matrix. We propose the intrinsic minimum average variance estimation and the intrinsic outer product of gradient method which fully exploit the geometric structure of the Riemannian manifold where the response resides. We present algorithms for our newly developed methods under the log-Euclidean metric and the log-Cholesky metric. The two metrics endow the manifold with a commutative Lie group structure that transforms our manifold model into a Euclidean one and helps us derive the consistency and asymptotic normality of estimators. Our methods are then naturally extended to the case allowing p = pn to diverge and the case of general Riemannian manifolds. Several simulation studies and an application to the New York taxi network data showcase the superiority of our proposals.

Key words and phrases: Central mean subspace, log-Cholesky metric, log-Euclidean metric, minimum average variance estimation, outer product of gradient, symmetric positive-definite matrix.