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

We consider two Riemannian geometries for the manifold M(p, m × n)
of all m × n matrices of rank p. The geometries are induced on M(p, m × n) by
viewing it as the base manifold of the submersion π : (M, N) → M NT, selecting
an adequate Riemannian metric on the total space, and turning π 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
onM(p, m × n) and to formulate the Riemannian Newton methods onM(p, m × n)
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.