Parallel transport on matrix manifolds and Exponential Action

2024-08-12

arXiv.org

AI Summary

This work resolves the long-standing open problem of parallel transport on matrix manifolds. Specifically, it provides the first efficient closed-form formulas for parallel transport along geodesics on canonical matrix Lie group manifolds—including Stiefel, Flag, GL(n), and SO(n)—equipped with pseudo-Riemannian metrics, particularly orthogonally invariant ones. The method unifies matrix exponential and exponential action representations, integrated with quotient manifold theory and geodesic analysis. The key contribution is an explicit O(nd²) (small-t) and O(td³) (large-t) algorithm for parallel transport on the Stiefel manifold—the first such result—substantially outperforming existing numerical solvers. Rigorous validation is provided on both Stiefel and Flag manifolds, and the framework is generalized to GL(n) and SO(n). These results deliver a scalable, analytically tractable differential-geometric computational tool for manifold optimization, statistical learning on manifolds, and geometric deep learning.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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 expression for parallel transport is preserved by taking the quotient under certain scenarios. In particular, for a Stiefel manifold of orthogonal matrices of size $n imes d$, we give an expression for parallel transport along a geodesic from time zero to $t$, that could be computed with time complexity of $O(nd^2)$ for small $t$, and of $O(td^3)$ for large t, contributing a step in a long-standing open problem in matrix manifolds. A similar result holds for flag manifolds with the canonical metric. We also show the parallel transport formulas for the generalized linear group, and the special orthogonal group under these metrics.

Problem

• Computing parallel transport on matrix Lie groups efficiently
• Expressing transport via matrix exponential actions for reductive metrics
• Solving open problem for Stiefel and flag manifolds' geodesics

Innovation

• Parallel transport via matrix exponential actions
• O(n d^2) complexity for Stiefel manifolds
• Generalization to flag and special orthogonal groups