An Efficient Algorithm for the Riemannian Logarithm on the Stiefel Manifold for a Family of Riemannian Metrics
Year: 2025
Type: article
Abstract: . Since the popularization of the Stiefel manifold for numerical applications in 1998 in a seminal paper from Edelman et al., it has been exhibited to be a key to solving many problems from optimization, statistics, and machine learning. In 2021, Hüper et al. proposed a one-parameter family of Riemannian metrics on the Stiefel manifold, subsuming the well-known Euclidean and canonical metrics. Since then, several methods have been proposed to obtain a candidate for the Riemannian logarithm given any metric from the family. Most of these methods are based on the shooting method or rely on optimization approaches. For the canonical metric, Zimmermann proposed in 2017 a particularly efficient method based on a pure matrix-algebraic approach. In this paper, we derive a generalization of this algorithm that works for the one-parameter family of Riemannian metrics. The algorithm is proposed in two versions, termed backward and forward, for which we prove that it conserves the local linear convergence previously exhibited in Zimmermann’s algorithm for the canonical metric. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/smataigne/StiefelLog.jl and in the supplementary materials ( supp-materials.zip [1.51MB]). [Formula: see text]
DOI: 10.1137/24m1647801
Authors: Simon Mataigne, Ralf Zimmermann, Nina Miolane
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4409216497