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

Marco Sutti

TL;DR

This work addresses the challenge of computing the Riemannian distance between points on the Stiefel manifold by introducing SSAF, a single shooting method that uses a canonical metric and an approximate Fréchet derivative of the geodesic. The method reformulates the endpoint problem as a root-finding problem for the geodesic endpoint, linearizes via a truncated Fréchet derivative of the matrix exponential, and solves a small Sylvester equation to update the geodesic parameters. A practical initial guess is obtained from a first-order exponential approximation and a projection onto the tangent space, enabling fast convergence. Numerical experiments show SSAF is competitive with or superior to several state-of-the-art approaches across large-scale instances, suggesting a favorable trade-off between accuracy and computational cost for manifold-valued distance computations in applications such as shape analysis and pattern recognition on St(n,p).

Abstract

This paper shows how to use the shooting method, a classical numerical algorithm for solving boundary value problems, to compute the Riemannian distance on the Stiefel manifold $ \mathrm{St}(n,p) $, the set of $ n \times p $ matrices with orthonormal columns. The proposed method is a shooting method in the sense of the classical shooting methods for solving boundary value problems; see, e.g., Stoer and Bulirsch, 1991. The main feature is that we provide an approximate formula for the Fréchet derivative of the geodesic involved in our shooting method. Numerical experiments demonstrate the algorithms' accuracy and performance. Comparisons with existing state-of-the-art algorithms for solving the same problem show that our method is competitive and even beats several algorithms in many cases.

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

TL;DR

This work addresses the challenge of computing the Riemannian distance between points on the Stiefel manifold by introducing SSAF, a single shooting method that uses a canonical metric and an approximate Fréchet derivative of the geodesic. The method reformulates the endpoint problem as a root-finding problem for the geodesic endpoint, linearizes via a truncated Fréchet derivative of the matrix exponential, and solves a small Sylvester equation to update the geodesic parameters. A practical initial guess is obtained from a first-order exponential approximation and a projection onto the tangent space, enabling fast convergence. Numerical experiments show SSAF is competitive with or superior to several state-of-the-art approaches across large-scale instances, suggesting a favorable trade-off between accuracy and computational cost for manifold-valued distance computations in applications such as shape analysis and pattern recognition on St(n,p).

Abstract

This paper shows how to use the shooting method, a classical numerical algorithm for solving boundary value problems, to compute the Riemannian distance on the Stiefel manifold , the set of matrices with orthonormal columns. The proposed method is a shooting method in the sense of the classical shooting methods for solving boundary value problems; see, e.g., Stoer and Bulirsch, 1991. The main feature is that we provide an approximate formula for the Fréchet derivative of the geodesic involved in our shooting method. Numerical experiments demonstrate the algorithms' accuracy and performance. Comparisons with existing state-of-the-art algorithms for solving the same problem show that our method is competitive and even beats several algorithms in many cases.
Paper Structure (17 sections, 61 equations, 4 figures, 6 tables, 2 algorithms)

This paper contains 17 sections, 61 equations, 4 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Applications and motivation
  4. Related works and other approaches
  5. Notation
  6. Outline of the paper
  7. Geometry of the Stiefel manifold
  8. Tangent spaces and projectors
  9. Geodesics, exponential mapping and logarithm mapping
  10. Problem statement
  11. A single shooting method with an approximate Fréchet derivative
  12. Linearization of the nonlinear matrix equation (\ref{['eq:nonlinear-eq']})
  13. The initial guess
  14. Numerical experiments and comparisons with other methods
  15. Comparisons with other state-of-the-art methods
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Illustration of the problem statement.
  • Figure 2: The Riemannian exponential map on the sphere.
  • Figure 3: Initial guess for the single shooting method on the Stiefel manifold.
  • Figure 4: Average computation times for Bryner's shooting method Bryner:2017, Zimmermann's matrix algebraic algorithm Zimmermann:2017, and our SSAF method on $\mathrm{St}(n,p)$. Left panel: plot corresponding to Table \ref{['tab:n5']}. Right panel: plot corresponding to Table \ref{['tab:n6']}.

Theorems & Definitions (2)

  • Remark 2.1
  • Remark 2.2