My work focuses on Brain-Computer Interfaces (BCIs), which aim to translate brain electrical signals into actionable commands. This is achieved using Electroencephalograms (EEGs) that measure the electrical activity of various brain regions. These EEG signals are transformed into covariance matrices, which offer insights into the connectivity between different brain areas. Covariance matrices have a unique structure: they are Symmetric Positive Definite (SPD) and thus belong to a Riemannian manifold.
In this talk, I will introduce BCI and motivate the use of the Riemannian geometry of SPD matrices in BCI. I will then describe the development of a Gaussian distribution on the manifold of SPD matrices. The goal is to model a set of SPD matrices with a probability distribution that inherently respects their Riemannian geometry. This is achieved by mapping a Euclidean multivariate Gaussian distribution onto the manifold of SPD matrices, resulting in a wrapped distribution.

I will present theoretical results on this wrapped Gaussian distribution, as well as methods for estimating its parameters from finite samples. Finally, I will demonstrate an application of wrapped Gaussians to the classification of SPD matrices