An efficient algorithm for the Riemannian logarithm on the Stiefel manifold for a family of Riemannian metrics
CoRR(2024)
摘要
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 solve 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.
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