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An new polar factor retraction on the Stiefel manifold with closed-form inverse

Rasmus Jensen, Ralf Zimmermann

TL;DR

This work addresses efficient retractions on the Stiefel manifold $St(n,p)$ and introduces a polar-light retraction that is second-order accurate under the Euclidean metric and possesses a closed-form inverse. The approach twists the classical polar-factor retraction and leverages a coordinate-chart construction to derive forward and inverse maps with tractable, closed-form expressions. The authors prove (theoretically) the retraction properties and demonstrate, through preliminary experiments, that the polar-light retraction better approximates geodesics than the full polar-factor retraction, especially for larger $p$, while keeping the inverse operation inexpensive. These features make the polar-light retraction attractive for log-map–heavy tasks such as Riemannian barycenters and manifold interpolation, and the accompanying Python code enables practical adoption and further acceleration opportunities such as Cayley-based approximations.

Abstract

Retractions are the workhorse in Riemannian computing applications, where computational efficiency is of the essence. This work introduces a new retraction on the compact Stiefel manifold of orthogonal frames. The retraction is second-order accurate under the Euclidean metric and features a closed-form inverse that can be efficiently computed. To the best of our knowledge, this is the first Stiefel retraction with both these properties. A variety of retractions is known on the Stiefel manifold, including the Riemannian exponential map, the polar factor retraction, the QR-retraction and the Cayley retraction, but none of them features a closed-form inverse. The only Stiefel retraction with closed-form inverse that we are aware of is based on quasi-geodesics, but this one is of first order.

An new polar factor retraction on the Stiefel manifold with closed-form inverse

TL;DR

This work addresses efficient retractions on the Stiefel manifold and introduces a polar-light retraction that is second-order accurate under the Euclidean metric and possesses a closed-form inverse. The approach twists the classical polar-factor retraction and leverages a coordinate-chart construction to derive forward and inverse maps with tractable, closed-form expressions. The authors prove (theoretically) the retraction properties and demonstrate, through preliminary experiments, that the polar-light retraction better approximates geodesics than the full polar-factor retraction, especially for larger , while keeping the inverse operation inexpensive. These features make the polar-light retraction attractive for log-map–heavy tasks such as Riemannian barycenters and manifold interpolation, and the accompanying Python code enables practical adoption and further acceleration opportunities such as Cayley-based approximations.

Abstract

Retractions are the workhorse in Riemannian computing applications, where computational efficiency is of the essence. This work introduces a new retraction on the compact Stiefel manifold of orthogonal frames. The retraction is second-order accurate under the Euclidean metric and features a closed-form inverse that can be efficiently computed. To the best of our knowledge, this is the first Stiefel retraction with both these properties. A variety of retractions is known on the Stiefel manifold, including the Riemannian exponential map, the polar factor retraction, the QR-retraction and the Cayley retraction, but none of them features a closed-form inverse. The only Stiefel retraction with closed-form inverse that we are aware of is based on quasi-geodesics, but this one is of first order.
Paper Structure (11 sections, 3 theorems, 30 equations, 1 figure, 2 tables)

This paper contains 11 sections, 3 theorems, 30 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. The classical Stiefel polar factor retraction
  4. The polar-light retraction
  5. Changing the coordinate center
  6. Retraction order
  7. Preliminary experimental results
  8. Accuracy
  9. Computation time
  10. Conclusions and future work
  11. Python code

Key Result

Lemma 3.1

Consider the special point $E= \in \operatorname{St}(n,p)$. The map is a coordinate chart on a (relative) open, path-connected neighborhood $\mathcal{B}\subset \operatorname{St}(n,p)$ around $E$.

Figures (1)

  • Figure 1: The figure shows the error $\|\operatorname{Exp}_{U0}(t\xi) - R_{U_0}(t\xi_R)\|_{F}$ between the Riemannian geodesic connecting $U_0$ and $U_1 = \operatorname{Exp}_{U_0}(t\xi)\vert_{t=1}$ and a retraction curve from $U_0$ to the same point $U_1 = R_{U_0}(t \xi_R)\vert_{t=1}$. As retractions, the polar factor retraction \ref{['eq:classic_PFR']} (PF, solid red line) and the polar-light retraction \ref{['eq:varphi_U_Onp2_efficient']} (PL, dashed black line) are considered.

Theorems & Definitions (5)

  • Lemma 3.1
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2