This WP will aim to develop novel DNNs for manifold-valued data/parameters. Several important applications involve training data and parameters that lie in a Riemannian space, e.g., graphs and point clouds [11]. This manifold structure should be reflected in the design of DNN architectures, and associated training losses. This results in network flows and optimisation problems in Riemannian spaces. Most existing solutions in this context have been proposed in an ad-hoc manner. In this WP, we will study DNNs on manifolds by drawing on Riemannian geometry, geometric measure theory and numerical linear algebra.