Sub Riemannian

Sub-Riemannian geometry extends Riemannian geometry by allowing for constraints on movement directions, leading to unique geometric structures and challenges in analysis. Current research focuses on developing algorithms for tasks like diffusion process simulation, gradient descent optimization in deep learning, and manifold learning, often leveraging adapted gradient descent methods and hypoelliptic diffusion processes. These advancements are proving valuable in diverse applications, including image processing (e.g., inpainting and enhancement), data analysis (e.g., dimension reduction and surface reconstruction), and statistical inference. The ability to model constrained movement offers powerful tools for analyzing complex data and systems where traditional Euclidean or Riemannian approaches are insufficient.