Newton's method: Optimization on manifolds

This lecture covers Newton's method for optimization on manifolds, focusing on exploiting second-order information to minimize a smooth function on a manifold. The instructor explains the choice of retraction, Riemannian metric, and initial point, and iterates algorithms to find the optimal solution. By utilizing second-order Taylor expansions and Hessians, a better search direction is determined. The lecture emphasizes the importance of choosing the right parameters for efficient convergence and discusses the drawbacks of Newton's method in terms of global behavior. Various fixes like trust-region methods and cubic regularization methods are presented to enhance the algorithm's performance.