# Parallel transport on matrix manifolds and Exponential Action

arXiv: https://arxiv.org/abs/2408.06054 · HTML: https://mlsanity.com/paper/2408.06054/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 44 references that are in the map.
- First posted on arXiv 12 Aug 2024.

## Builds on (most cited first, 7 of 7)

- [Optimization on flag manifolds](https://mlsanity.com/paper/1907.00949/) (2019)
- [Riemannian conjugate gradient methods: General framework and specific algorithms with convergence analyses](https://mlsanity.com/paper/2112.02572/) (2021)
- [Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds](https://mlsanity.com/paper/2103.13327/) (2021)
- [Fun with Flags: Robust Principal Directions via Flag Manifolds](https://mlsanity.com/paper/2401.04071/) (2024)
- [Geometric Optimization Methods for Adaptive Filtering](https://mlsanity.com/paper/1305.1886/) (2013)
- [Extremal curves on Stiefel and Grassmann manifolds](https://mlsanity.com/paper/1809.07013/) (2018)
- [Curvatures of Stiefel manifolds with deformation metrics](https://mlsanity.com/paper/2105.01834/) (2021)

## Built on by (most cited first, 1 of 1)

- [Rolled Gaussian process models for curves on manifolds](https://mlsanity.com/paper/2503.21980/) (2025)

## Abstract (arXiv)

We express parallel transport for several common matrix Lie groups with a family of pseudo-Riemannian metrics in terms of matrix exponential and exponential actions. The metrics are constructed from a deformation of a bi-invariant metric and are naturally reductive. There is a similar picture for homogeneous spaces when taking quotients satisfying a general condition. In particular, for a Stiefel manifold of orthogonal matrices of size n× d, we give an expression for parallel transport along a geodesic from time zero to t, that could be computed with time complexity of O(n d2) for small t, and of O(td3) for large t, contributing a step in a long-standing open problem in matrix manifolds. A similar result holds for it flag manifolds with the canonical metric. We also show the parallel transport formulas for the it general linear group and the it special orthogonal group under these metrics.