Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries

We consider two Riemannian geometries for the manifold of all matrices of rank . The geometries are induced on by viewing it as the base manifold of the submersion , selecting an adequate Riemannian metric on the total space, and turning into a Riemannian submersion. The theory of Riemannian submersions, an important tool in Riemannian geometry, makes it possible to obtain expressions for fundamental geometric objects on and to formulate the Riemannian Newton methods on induced by these two geometries. The Riemannian Newton methods admit a stronger and more streamlined convergence analysis than the Euclidean counterpart, and the computational overhead due to the Riemannian geometric machinery is shown to be mild. Potential applications include low-rank matrix completion and other low-rank matrix approximation problems.