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

We consider two Riemannian geometries for the manifold $${\mathcal{M }(p,m\times n)}$$ M ( p , m × n ) of all $$m\times n$$ m × n matrices of rank $$p$$ p . The geometries are induced on $${\mathcal{M }(p,m\times n)}$$ M ( p , m × n ) by viewing it as the base manifold of the submersion $$\pi :(M,N)\mapsto MN^\mathrm{T}$$ π : ( M , N ) ↦ M N T , selecting an adequate Riemannian metric on the total space, and turning $$\pi $$ 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 on $${\mathcal{M }(p,m\times n)}$$ M ( p , m × n ) and to formulate the Riemannian Newton methods on $${\mathcal{M }(p,m\times n)}$$ M ( 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. Copyright Springer-Verlag Berlin Heidelberg 2014