# Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics

arXiv: https://arxiv.org/abs/2408.07072 · HTML: https://mlsanity.com/paper/2408.07072/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 25 references that are in the map.
- First posted on arXiv 25 Jul 2024.

## Builds on (most cited first, 6 of 6)

- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Orthogonal Weight Normalization: Solution to Optimization over Multiple Dependent Stiefel Manifolds in Deep Neural Networks](https://mlsanity.com/paper/1709.06079/) (2017)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Optimization flows landing on the Stiefel manifold](https://mlsanity.com/paper/2202.09058/) (2022)
- [Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds](https://mlsanity.com/paper/2103.13327/) (2021)
- [The ultimate upper bound on the injectivity radius of the Stiefel manifold](https://mlsanity.com/paper/2403.02079/) (2024)

## Built on by (most cited first, 2 of 2)

- [Near-optimal Rank Adaptive Inference of High Dimensional Matrices](https://mlsanity.com/paper/2510.08117/) (2025)
- [Stochastic Optimization of Tree Tensor Networks](https://mlsanity.com/paper/2609.00870/) (2026)

## Abstract (arXiv)

We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by H"uper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications.