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We consider two Riemannian geometries for the manifold $\mathcal{M}(p,m\timesn)$ 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$\pi:(M,N)\mapsto MN^T$, selecting an adequate Riemannian metric on the totalspace, and turning $\pi$ into a Riemannian submersion. The theory of Riemanniansubmersions, an important tool in Riemannian geometry, makes it possible toobtain expressions for fundamental geometric objects on $\mathcal{M}(p,m\timesn)$ and to formulate the Riemannian Newton methods on $\mathcal{M}(p,m\timesn)$ induced by these two geometries. The Riemannian Newton methods admit astronger and more streamlined convergence analysis than the Euclideancounterpart, and the computational overhead due to the Riemannian geometricmachinery is shown to be mild. Potential applications include low-rank matrixcompletion and other low-rank matrix approximation problems.

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