Riemannian Optimization Framework on the Generalized Quaternionic Stiefel Manifold

TL;DR

This paper develops a Riemannian optimization framework on the generalized quaternionic Stiefel manifold, providing geometric insights and practical algorithms for problems like the generalized quaternionic eigenvalue problem.

Contribution

It introduces the generalized quaternionic Stiefel manifold, derives explicit geometric quantities, and applies Riemannian optimization to solve related eigenvalue problems.

Findings

Explicit geometric expressions for the manifold are derived.

The framework is validated through a numerical example solving a quaternionic eigenvalue problem.

The approach demonstrates the viability of Riemannian optimization on this new manifold.

Abstract

This paper introduces the generalized quaternionic Stiefel manifold and studies its geometry for Riemannian optimization. We clarify its relationships with existing manifolds, especially the real generalized Stiefel manifold and the quaternionic Stiefel manifold, and derive explicit expressions for several geometric quantities on the proposed manifold. In particular, the generalized quaternionic Stiefel manifold is regarded as a real Riemannian manifold, and expressions for the tangent space, normal space, the orthogonal projection onto the tangent space, a retraction, and a vector transport on this manifold are derived. As an application, the generalized quaternionic eigenvalue problem is formulated as an optimization problem on this manifold, and a numerical example is solved by Riemannian optimization methods to demonstrate the viability of the proposed framework.