A generalized power iteration method for solving quadratic problem on the Stiefel manifold

TL;DR

This paper introduces a generalized power iteration method for solving quadratic problems on the Stiefel manifold, providing theoretical convergence analysis and demonstrating empirical efficiency for related problems.

Contribution

A novel generalized power iteration method is proposed for quadratic problems on the Stiefel manifold, with theoretical analysis and applications to orthogonal least squares regression and unbalanced orthogonal Procrustes problem.

Findings

The GPI method converges theoretically.

The approach is empirically efficient.

Applications to UOPP show promising results.

Abstract

In this paper, we first propose a novel generalized power iteration method (GPI) to solve the quadratic problem on the Stiefel manifold (QPSM) as min_{W^TW=I}Tr(W^TAW-2W^TB) along with the theoretical analysis. Accordingly, its special case known as the orthogonal least square regression (OLSR) is under further investigation. Based on the aforementioned studies, we then cast major focus on solving the unbalanced orthogonal procrustes problem (UOPP). As a result, not only a general convergent algorithm is derived theoretically but the efficiency of the proposed approach is verified empirically as well.