Geometric design of the tangent term in landing algorithms for orthogonality constraints

TL;DR

This paper introduces a new family of metrics for the set of full-rank matrices, extending the $eta$-metric on the Stiefel manifold, to improve landing algorithms for orthogonality-constrained optimization.

Contribution

It proposes a novel family of metrics tailored for full-rank matrices, enhancing the geometric framework for orthogonality-constrained optimization problems.

Findings

New metrics extend the $eta$-metric to full-rank matrices.

Application to landing algorithms improves optimization under orthogonality constraints.

Provides a geometric foundation for better algorithm design.

Abstract

We propose a family a metrics over the set of full-rank $n\times p$ real matrices, and apply them to the landing framework for optimization under orthogonality constraints. The family of metrics we propose is a natural extension of the $\beta$-metric, defined on the Stiefel manifold.