Variations of Orthonormal Basis Matrices of Subspaces

TL;DR

This paper investigates how orthonormal basis matrices of subspaces vary under specific normalization constraints, providing bounds useful for analyzing convergence in multi-view subspace learning models.

Contribution

It derives bounds on the variation of orthonormal basis matrices when the subspace changes under a positive semi-definiteness constraint on $X^{ m T}D$, aiding convergence analysis.

Findings

Bounds on the change of basis matrices derived

Results applicable to convergence analysis of NEPv methods

Enhances understanding of basis matrix stability under constraints

Abstract

An orthonormal basis matrix $X$ of a subspace ${\cal X}$ is known not to be unique, unless there are some kinds of normalization requirements. One of them is to require that $X^{\rm T}D$ is positive semi-definite, where $D$ is a constant matrix of apt size. It is a natural one in multi-view subspace learning models in which $X$ serves as a projection matrix and is determined by a maximization problem over the Stiefel manifold whose objective function contains and increases with tr$(X^{\rm T}D)$. This paper is concerned with bounding the change in orthonormal basis matrix $X$ as subspace ${\cal X}$ varies under the requirement that $X^{\rm T}D$ stays positive semi-definite. The results are useful in convergence analysis of the NEPv approach (nonlinear eigenvalue problem with eigenvector dependency) to solve the maximization problem.