On the Latent Information Geometry of the Grassmann Manifold

Modeling linear subspaces and relations among them naturally arises in several applications in signal processing, computer vision, and system identification. In this paper, we investigate the latent information geometry of deep generative models that output linear subspaces. Such subspaces are members of the Grassmann manifold, which we model with a matrix Bingham distribution as a likelihood. We derive the Fisher-Rao metric on the statistical manifold of the matrix Bingham parameters, and propose pulling this back to the latent space to achieve uncertainty-aware and identifiable latent representations. We provide numerical results assessing the meaningfulness of the achieved latent subspace representations on a relevant vehicular wireless communications scenario.