Wrapped Distributions on homogeneous Riemannian manifolds

Probability distributions play a fundamental role in statistical data analysis where, for continuous data, the dominant assumption is to consider random variables in Euclidean space. However, the Euclidean assumption is not appropriate for some data types and this motivates the development and study of distributions in non-Euclidean spaces. Notable examples include directional statistics (see [21]), in which observations typically lie on a sphere, and data that are expressed as tensors, such as covariance matrices ([32, 35, 36]) and data structures which arise in image and signal processing applications (see [2, 3]). Furthermore, in recent years, latent variable models have been shown to offer superior performance when the parameters are modelled in non-Euclidean spaces. Variational autoencoders (see [16, 27, 40, 23, 30]) and latent space network models (see [14, 19, 41, 24, 22]) offer two pertinent examples where it is most typical for non-Euclidean latent variables to be modelled in spherical or hyperbolic space.