Riemannian preconditioning

TL;DR

This paper introduces a Riemannian preconditioning approach that leverages the connection between quadratic programming and Riemannian gradient methods to improve optimization on quotient manifolds, especially for matrix problems.

Contribution

It proposes a novel metric selection method for Riemannian optimization, enhancing efficiency in quadratic problems with orthogonality or rank constraints.

Findings

Effective in quadratic optimization with orthogonality constraints

Improves efficiency of Riemannian gradient methods

Applicable to a wide range of matrix manifold problems

Abstract

This paper exploits a basic connection between sequential quadratic programming and Riemannian gradient optimization to address the general question of selecting a metric in Riemannian optimization, in particular when the Riemannian structure is sought on a quotient manifold. The proposed method is shown to be particularly insightful and efficient in quadratic optimization with orthogonality and/or rank constraints, which covers most current applications of Riemannian optimization in matrix manifolds.