Steepest descent algorithm on orthogonal Stiefel manifolds

TL;DR

This paper develops an explicit steepest descent algorithm for optimization on orthogonal Stiefel manifolds, utilizing ambient coordinates and analyzing the influence of submatrix choices on the algorithm's behavior.

Contribution

It provides a novel explicit construction of local coordinates and a steepest descent algorithm on orthogonal Stiefel manifolds, including conditions for critical points and analysis of submatrix effects.

Findings

Explicit local coordinate description for Stiefel manifolds.

Conditions for critical points of functions on Stiefel manifolds.

Implementation of steepest descent algorithm using ambient coordinates.

Abstract

Considering orthogonal Stiefel manifolds as constraint manifolds, we give an explicit description of a set of local coordinates that also generate a basis for the tangent space in any point of the orthogonal Stiefel manifolds. We show how this construction depends on the choice of a submatrix of full rank. Embedding a gradient vector field on an orthogonal Stiefel manifold in the ambient space, we give explicit necessary and sufficient conditions for a critical point of a cost function defined on such manifolds. We explicitly describe the steepest descent algorithm on the orthogonal Stiefel manifold using the ambient coordinates and not the local coordinates of the manifold. We point out the dependence of the recurrence sequence that defines the algorithm on the choice of a full rank submatrix. We illustrate the algorithm in the case of Brockett cost functions.