Kernels on the Riemannian manifold of symmetric positive definite matrices

This thesis presents a formalization of kernel methods for Riemannian manifolds, with a focus on the manifold of symmetric positive definite (SPD) matrices. The work introduces the concept of Riemannian linear features, a novel approach to linearizing the manifold that incorporates the linear structure of the tangent space given by the Riemannian metric. Two general classes of kernels are described and compared for commonly used Riemannian metrics on SPD matrices: canonical kernels and metric space kernels. The thesis provides a comprehensive theoretical framework, including a detailed exploration of differential and Riemannian geometry, metric spaces, and distance geometry. It then applies this framework to develop new formalizations of the Riemannian manifold of positive definite matrices, describing standard metrics and showing how the new concepts are incorporated into each metric. The performance of the proposed kernel methods is evaluated through two comparisons: classification and dimensionality reduction. These are validated using a two-class motor imagery task with electroencephalography (EEG) recordings, employing a cross-session within-subject leave-one-session-out paradigm. This work contributes to the field by offering a unified formalization of kernel methods for Riemannian manifolds and SPD matrices in particular. It provides insights into the construction and application of kernels in this framework, opening up new avenues for research in areas such as Brain-Computer Interfaces and other domains where SPD matrices can be extracted as features for machine learning methods.