Paper ID sheet UCL-INMA-2012.05

Title

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

Authors

P.-A. Absil, Luca Amodei, Gilles Meyer

Abstract

We consider two Riemannian geometries for the manifold $M(p,m\times n)$ of all $m\times n$ matrices of rank $p$. The geometries are induced on $M(p,m\times n)$ by viewing it as the base manifold of the submersion $\pi:(M,N)\mapsto MN^\T$, selecting an adequate Riemannian metric on the 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 $M(p,m\times n)$ and to formulate the Riemannian Newton methods on $M(p,m\times n)$ 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.

Key words

fixed-rank matrices; manifold; differential geometry; Riemannian geometry; Riemannian submersion; Levi-Civita connection; Riemannian connection; Riemannian exponential map; geodesics; Newton's method

Status

Computational Statistics, June 2014, Volume 29, Issue 3-4, pp 569-590

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• Publisher's version: doi:10.1007/s00180-013-0441-6
• Preprint

BibTeX entry

@ARTICLE{AbsAmoMey2014,
author = "P.-A. Absil and Luca Amodei and Gilles Meyer",
title = "Two {Newton} methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries",
journal = "Computational Statistics",
pages = "569-590",
year = 2014,
volume = 29,
number = "3-4",
doi = "10.1007/s00180-013-0441-6",
}