Riemannian gradient and Levi-Civita connection for fixed-rank matrices

TL;DR

This paper derives formulas for Riemannian gradient and Levi-Civita connection on fixed-rank matrix manifolds using metrics based on Stiefel manifolds, aiding optimization in low-rank matrix problems.

Contribution

It introduces explicit formulas for Riemannian gradient and Levi-Civita connection tailored for fixed-rank matrices with nonconstant metrics.

Findings

Formulas for Riemannian gradient on fixed-rank matrices

Formulas for Levi-Civita connection on fixed-rank matrices

Metrics based on nonconstant metrics on Stiefel manifolds

Abstract

We provide formulas for Riemannian gradient and Levi-Civita connection for a family of metrics on fixed-rank matrix manifolds, based on nonconstant metrics on Stiefel manifolds.