  We are interested in the challenging problem of modelling densities on
Riemannian manifolds with a known symmetry group using normalising flows. This
has many potential applications in physical sciences such as molecular dynamics
and quantum simulations. In this work we combine ideas from implicit neural
layers and optimal transport theory to propose a generalisation of existing
work on exponential map flows, Implicit Riemannian Concave Potential Maps,
IRCPMs. IRCPMs have some nice properties such as simplicity of incorporating
symmetries and are less expensive than ODE-flows. We provide an initial
theoretical analysis of its properties and layout sufficient conditions for
stable optimisation. Finally, we illustrate the properties of IRCPMs with
density estimation experiments on tori and spheres.
