Geometric Learning: Leveraging differential geometry for learning and control

In this thesis, we concentrate on advancing high-level behavioral control policies for robotic systems within the framework of Dynamical Systems (DS). Throughout the course of this research, a unifying thread weaving through diverse fields emerges, and that is the fundamental role played by differential geometry. This study delves into various realms of this mathematical framework, with three distinct projects at its core. The first work revolves around graph Laplacian-based embedding space reconstruction, followed by an exploration of chart-based geometry in the second project. The third project shifts its focus towards harmonic analysis in non-Euclidean spaces. These facets of differential geometry, while seemingly distinct, converge in their practical application within the realm of robotics, specifically in the domain of Dynamical Systems (DS) based robot motion generation. The first two projects employ differential geometry tools for the purpose of learning and clustering DS on Euclidean spaces, whereas the third project ventures into the potential domain of learning DS on non-Euclidean spaces, otherwise known as manifolds. Our investigation into a more sophisticated geometry-based formalism is directed not only at enhancing the expressivity and complexity of DS policies for navigating intricate and dynamic real-world scenarios but also at favouring a more profound comprehension of the practical application of rather abstract mathematical concepts in the field of robotics and machine learning.