A thorough study of Riemannian Newton's Method

This work presents a thorough numerical study of Riemannian Newton's Method (RNM) for optimization problems, with a focus on the Grassmannian and on the Stiefel manifold. We compare the Riemannian formulation of Newton's Method with its classical Euclidean counterpart based on Lagrange multipliers by applying both approaches to the important and challenging Hartree--Fock energy minimization problem from Quantum Chemistry. Experiments on a dataset of 125 molecules show that the Riemannian approaches achieve higher convergence rates, require fewer iterations, and exhibit greater robustness to the choice of initial guess. In this work we also analyze the numerical issues that arise from using Newton's Method on the total manifold when the cost function is defined on the quotient manifold. We investigate the performance of a modified RNM in which we ignore the small eigenvalues of the Hessian and the results indicate that this modified method is stable and performs on par with the RNM on the quotient manifold.