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

2012/09/01 by Pierre-Antoine Absil, Absil, P. -A., Luca Amodei +3 · 1 citation
Computer Science · Engineering · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Statistical and numerical algorithms

paper · doi:10.48550/arxiv.1209.0068

openalex publication_date 2012/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Low-rank optimization with trace norm penalty

Cited by

• A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

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