Symmetric Spaces of Groups over Involutive Algebras for Graph Embeddings

Symmetric spaces such as the Siegel upper half-space and the Siegel disk domain have proven promising for graph embedding. Recently, they have also been proposed as representation spaces for discriminative neural networks. In this paper, we are interested in generalizations of Siegel spaces for representation learning. In particular, we focus on symmetric spaces associated with symplectic groups over involutive algebras, which have recently been introduced as generalizations of the real symplectic group. Although these symmetric spaces offer the capacity to capture rich geometric structures, core components required for practical implementation of neural networks on these space are missing, making the task of learning such parametric models difficult. To this end, we further develop the theory of symplectic groups over involutive algebras by deriving closed-form formulas for some missing components, i.e., automorphisms and their inverses. We consider realizations of the studied spaces, and demonstrate their effectiveness for graph tasks.