A Numerical Method for Computing Geodesics on the Stiefel Manifold

Core Concepts

This paper presents a single shooting method with an approximate Fréchet derivative for efficiently computing geodesics on the Stiefel manifold.

Abstract

The paper focuses on the problem of computing the Riemannian distance between two points on the Stiefel manifold, which is equivalent to finding the tangent vector with the shortest length that connects the two points.
Key highlights:

The authors use the classical shooting method, a numerical algorithm for solving boundary value problems, to compute geodesics on the Stiefel manifold.
They introduce an approximate Fréchet derivative to linearize the nonlinear matrix equation, leading to an efficient linear matrix equation that can be solved.
Extensive numerical experiments demonstrate the accuracy and performance of the proposed algorithm, showing it is competitive with and even outperforms several existing state-of-the-art methods.
The authors also provide an effective initialization strategy for the shooting method to ensure good convergence.
The proposed method is applicable to various domains that involve data on the Stiefel manifold, such as numerical optimization, imaging, and signal processing.

Stats

The Stiefel manifold St(n, p) is the set of n × p matrices with orthonormal columns.
The dimension of the Stiefel manifold is dim(St(n, p)) = np - 1/2 p(p + 1).

Quotes

"No explicit formula is known for computing the distance on St(n, p), one has to resort to numerical methods."
"Shooting methods are not the only option to solve the endpoint geodesic problem; many other numerical algorithms have been proposed."

Key Insights Distilled From

A single shooting method with approximate Fréchet derivative for computing geodesics on the Stiefel manifold

by Marco Sutti at arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.04089.pdf

Deeper Inquiries

How can the proposed single shooting method be extended to handle more general Riemannian metrics on the Stiefel manifold beyond the canonical metric

The proposed single shooting method can be extended to handle more general Riemannian metrics on the Stiefel manifold by incorporating the metric tensor into the geodesic equation. In the context of Riemannian geometry, the geodesic equation is typically defined in terms of the Levi-Civita connection, which is determined by the metric tensor. By modifying the geodesic equation to include the metric tensor, the single shooting method can be adapted to work with different metrics on the Stiefel manifold.

What are the theoretical convergence guarantees of the single shooting method with the approximate Fréchet derivative, and how do they compare to the exact Fréchet derivative approach

The theoretical convergence guarantees of the single shooting method with the approximate Fréchet derivative can be analyzed using standard convergence analysis techniques for numerical algorithms. The convergence properties of the method depend on factors such as the smoothness of the objective function, the choice of step size in the iterative process, and the accuracy of the approximation of the Fréchet derivative.
Compared to the exact Fréchet derivative approach, the approximate Fréchet derivative method may have slightly slower convergence rates due to the approximation error introduced. However, in practice, the approximate Fréchet derivative approach can still achieve convergence to a high degree of accuracy and is computationally more efficient, especially for large-scale problems where computing the exact Fréchet derivative may be prohibitively expensive.

Can the ideas behind the efficient initialization strategy be applied to other numerical methods for computing geodesics on Riemannian manifolds beyond the Stiefel manifold

The efficient initialization strategy used in the proposed single shooting method for computing geodesics on the Stiefel manifold can be applied to other numerical methods for computing geodesics on Riemannian manifolds beyond the Stiefel manifold. By utilizing a first-order approximation of the matrix exponential and projecting the initial guess onto the tangent space, the initialization strategy can help improve the convergence properties of various numerical methods for solving geodesic problems on different Riemannian manifolds.
The key idea behind the initialization strategy is to provide a good starting point for the iterative optimization process, which can help accelerate convergence and improve the overall efficiency of the numerical method. This strategy can be adapted and implemented in a similar manner for other numerical algorithms that involve solving geodesic equations on different Riemannian manifolds.