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• W2078248112 abstract "We consider two Riemannian geometries for the manifold $${mathcal{M }(p,mtimes n)}$$ of all $$mtimes n$$ matrices of rank $$p$$ . The geometries are induced on $${mathcal{M }(p,mtimes n)}$$ 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 $${mathcal{M }(p,mtimes n)}$$ and to formulate the Riemannian Newton methods on $${mathcal{M }(p,mtimes n)}$$ 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." @default.
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• W2078248112 date "2013-08-14" @default.
• W2078248112 modified "2023-10-02" @default.
• W2078248112 title "Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries" @default.
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• W2078248112 doi "https://doi.org/10.1007/s00180-013-0441-6" @default.
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