Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries
We consider two Riemannian geometries for the manifold \({\mathcal{M }(p,m\times n)}\) of all \(m\times n\) matrices of rank \(p\). The geometries are induced on \({\mathcal{M }(p,m\times n)}\) by viewing it as the base manifold of the submersion \(\pi :(M,N)\mapsto MN^\mathrm{T}\), selecting an adequate Riemannian metric on the total space, and turning \(\pi \) 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 \({\mathcal{M }(p,m\times n)}\) and to formulate the Riemannian Newton methods on \({\mathcal{M }(p,m\times n)}\) 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.
riemannian submersion fixed-rank manifold levi-civita connection riemannian connection riemannian exponential map geodesics