# Stochastic Riemannian Geometry with Applications toData Modelling

## Project Details

### Layman's description

Imagine a sphere that has been embedded into to three dimensional space with radius one. All points on the sphere can be written in three dimensional space, but they can also be written in spherical coordinates meaning that the points in the three dimensional space can be represented in a latent two dimensional space via a parametrization of the sphere. A classical problem in data modelling is thus to learn the mapping from the parametrizing coordinates to the three dimensional coordinates based only on observed data.

Assume that we have observed data points, which all are on the northern hemisphere. Then the latent representation of these points can be learned using e.g. a Variational Autoencoder (VAE), which is an example of a deep generative model from machine learning. In this way a two dimensional latent representation of the data can be learned as well as the geometry of the data. This representation has been seen to be relatively accurate. However, given all the points are on the northern hemisphere the VAE has obviously difficulties in learning the geometry of the data in areas with no or scarce training data and correspondingly high uncertainty. The problem using VAE is that it will not necessarily discard the southern hemisphere in the data analysis - meaning that shortest curves (known as geodesics) might go through regions where there has been no training data. We therefore need to use a model that takes the uncertainty of the data observations into account!

Bayesian inference provides a way of taking the uncertainty of the data into account in the modelling phase such that the learned latent representation identifies in which regions there has been a higher number of data-points and thus high certainty, and in which regions there has been scarce or no training data and therefore high uncertainty. In the example of the sphere the Bayesian model will take into account the certainty in the northern hemisphere and include the uncertainty in the southern hemisphere, which will ensure that e.g. shortest curves will avoid regions with no training data.

A sphere is an example of a more general concept known as a Riemannian manifold. The example is a simple case of extracting features from high-dimensional data forming a low dimensional latent representation of the data. In general it is often hypothesised that the high-dimensional data lies on a curved and topologically non-trivial manifold.

For general high-dimensional data it has been observed that using Bayesian inference performs better when learning the geometry of the data compared to using deterministic methods such as VAE's. However, in the Bayesian set-up the mapping from the latent representation into the high dimensional space will mathematically be a (smooth) stochastic process (with probability 1). In this way the model will not learn only one deterministic Riemannian manifold, but a distribution of Riemannian manifolds known as a stochastic Riemannian manifold.

Riemannian geometry has already been used in a wide number of applications, e.g. image analysis and biomedical engineering such as diffusion MRI. The objective of the Ph.D. project is to develop the mathematical theory necessary for the study of stochastic Riemannian manifolds in order to interact with data under a stochastic metric as well as developing algorithms to study the behaviour of the stochastic Riemannian manifolds and their properties in data modelling.
Status Active 15/12/2022 → 14/12/2025