ncts

In this talk, I will showcase several fundamental ideas and tools needed to perform numerical optimization on matrix manifolds. We will start off with a brief overview of the steepest descent method, a classical optimization algorithm in Euclidean space. Then, we will introduce the main tools from Riemannian geometry that allow us to extend the steepest descent to matrix manifolds, with particular emphasis on the retraction mapping, and present some numerical examples. To keep the presentation more concrete, we will fix our ideas on the Stiefel manifold, but the concepts discussed herein would remain valid for any embedded submanifold. If time permits, I will present an extension of nonlinear multigrid to Riemannian manifolds. In this new multilevel algorithm, the optimization variable is constrained to the Riemannian manifold of fixed-rank matrices, allowing to keep the ranks (and thus the computational complexity) fixed throughout the iterations.