A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric

TL;DR

This paper introduces a matrix-algebraic numerical algorithm for computing the Riemannian logarithm on the Stiefel manifold under the canonical metric, offering a convergence guarantee and an alternative to optimization-based methods.

Contribution

It presents a novel matrix-algebraic approach for the Riemannian logarithm on the Stiefel manifold, differing from existing optimization-based techniques.

Findings

Algorithm converges locally with linear rate

Provides a purely matrix-algebraic method

Offers an alternative to optimization-based approaches

Abstract

We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the existing optimization-based approach, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm converges locally and exhibits a linear rate of convergence.