Wrapped Distributions on homogeneous Riemannian manifolds

TL;DR

This paper introduces a flexible framework for constructing probability distributions on Riemannian manifolds using area-preserving maps and isometries, enabling applications in machine learning models like autoencoders.

Contribution

It provides a general method for creating distributions on manifolds with controllable properties, suitable for sampling and integration into latent variable models.

Findings

Validated distributions within variational autoencoders

Demonstrated applicability to latent space network models

Proposed future research directions

Abstract

We provide a general framework for constructing probability distributions on Riemannian manifolds, taking advantage of area-preserving maps and isometries. Control over distributions' properties, such as parameters, symmetry and modality yield a family of flexible distributions that are straightforward to sample from, suitable for use within Monte Carlo algorithms and latent variable models, such as autoencoders. As an illustration, we empirically validate our approach by utilizing our proposed distributions within a variational autoencoder and a latent space network model. Finally, we take advantage of the generalized description of this framework to posit questions for future work.