Implicit Riemannian Concave Potential Maps
This work addresses density estimation challenges in physical sciences such as molecular dynamics and quantum simulations, representing an incremental improvement over existing exponential map flows.
The paper tackles the problem of modeling densities on Riemannian manifolds with known symmetry groups using normalizing flows, proposing Implicit Riemannian Concave Potential Maps (IRCPMs) as a generalization of exponential map flows. The method demonstrates properties like simpler symmetry incorporation and lower computational cost than ODE-flows, with experimental validation on tori and spheres.
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.