Implicit Riemannian Concave Potential Maps

TL;DR

This paper introduces Implicit Riemannian Concave Potential Maps (IRCPMs), a novel method for modeling densities on Riemannian manifolds that leverages implicit neural layers and optimal transport, offering symmetry incorporation and computational efficiency.

Contribution

The paper proposes IRCPMs, a new class of flows that generalize exponential map flows for Riemannian manifolds, with theoretical analysis and practical density estimation experiments.

Findings

IRCPMs effectively model densities on tori and spheres.

IRCPMs are computationally less expensive than ODE-based flows.

Theoretical conditions for stable optimization of IRCPMs are provided.

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.