Gradient Descent

This lecture covers optimization on manifolds, focusing on a family of gradient descent methods to minimize a smooth function on a manifold. The algorithm template for (Riemannian) gradient descent is discussed, emphasizing the aim to guarantee small gradients. The lecture also delves into Taylor expansions, first-order, and the concept of retraction and Riemannian metric on the manifold. The instructor explains the algorithm's steps and highlights the importance of ensuring small gradients for convergence. The lecture concludes with insights on global convergence and the regularity assumption, particularly the Lipschitz-type assumption. Various mathematical expressions and theorems are presented to support the concepts discussed.