Visualizing Riemannian data with Rie-SNE

TL;DR

This paper introduces Rie-SNE, an extension of t-SNE for visualizing data on Riemannian manifolds, accounting for geometry and enabling manifold-to-manifold mappings.

Contribution

It develops a Riemannian version of SNE using diffusion-based assumptions and efficient approximations for manifold data visualization.

Findings

Effective visualization of manifold data

Mapping between different Riemannian manifolds

Maintains geometric fidelity in low-dimensional embeddings

Abstract

Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.