  Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD)
matrices and Grassmann manifolds, appear in many applications. Recently, by
applying the theory of gyrogroups and gyrovector spaces that is a powerful
framework for studying hyperbolic geometry, some works have attempted to build
principled generalizations of Euclidean neural networks on matrix manifolds.
However, due to the lack of many concepts in gyrovector spaces for the
considered manifolds, e.g., the inner product and gyroangles, techniques and
mathematical tools provided by these works are still limited compared to those
developed for studying hyperbolic geometry. In this paper, we generalize some
notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new
models and layers for building neural networks on these manifolds. We show the
effectiveness of our approach in two applications, i.e., human action
recognition and knowledge graph completion.
