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Riemannian Optimization Framework on the Generalized Quaternionic Stiefel Manifold

Hiroyuki Sato

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.

Riemannian Optimization Framework on the Generalized Quaternionic Stiefel 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.
Paper Structure (5 sections, 10 theorems, 55 equations, 2 figures)

This paper contains 5 sections, 10 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and review of quaternions
  3. Generalized quaternionic Stiefel manifold
  4. Application to generalized quaternionic eigenvalue problem
  5. Concluding remarks

Key Result

Proposition 3.1

The generalized quaternionic Stiefel manifold $\mathop{\mathrm{St}}\nolimits_G(p,\mathbb{H}^n)$ is a $p(4n - 2p + 1)$-dimensional real submanifold of $\mathbb{H}^{n \times p}$. Strictly speaking, $\psi(\mathop{\mathrm{St}}\nolimits_G(p,\mathbb{H}^n))$ is an embedded submanifold of $\psi(\mathbb{H}^{

Figures (2)

  • Figure 1: Riemannian gradient norm versus iterations.
  • Figure 2: Riemannian gradient norm versus elapsed time.

Theorems & Definitions (21)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.4
  • proof
  • ...and 11 more