Bounded Domains for Symplectic Groups and Applications to Representation Learning

Neural networks on the Siegel disk domain have recently been introduced as an effective tool for addressing a number of machine learning tasks. These networks are built upon the complex structure of the Siegel disk domain and its connection with the real symplectic group. In this paper, we study generalizations of the real symplectic group referred to as symplectic groups over involutive algebras, and propose a class of upper half-space and precompact models associated with these groups. We derive closed-form formulas for some mappings required to implement neural networks on the precompact model, i.e., automorphisms and their inverses. We give explicit expressions for these mappings on a class of bounded domains, and investigate their representation power on the tasks of radar clutter classification, action recognition, graph reconstruction, and node classification. Our experimental results demonstrate the efficacy of the proposed approach.