Implicit Riemannian Concave Potential Maps

Danilo Jimenez Rezende, Sebastien Racaniere

Abstract:

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.