# Diffeomorphic random sampling using optimal information transport

## TL;DR

This paper introduces a diffeomorphic sampling algorithm on Riemannian manifolds based on optimal information transport, offering a semi-explicit alternative to optimal mass transport and MCMC methods.

## Contribution

It develops a novel sampling method leveraging the Fisher-Rao metric and information geometry, avoiding complex nonlinear equations of traditional OMT.

## Key findings

- Semi-explicit formulation of the sampling algorithm.
- Potential advantages over MCMC for large sample sizes.
- Connections between Fisher-Rao metric and diffeomorphism groups.

## Abstract

In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)---an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. The resulting sampling algorithm is a promising alternative to OMT, in particular as our formulation is semi-explicit, free of the nonlinear Monge--Ampere equation. Compared to Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when a large number of samples from a low dimensional nonuniform distribution is needed.