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

TL;DR

This paper introduces a novel second-order accurate retraction on the Stiefel manifold with a closed-form inverse, enhancing computational efficiency in Riemannian optimization.

Contribution

It presents the first second-order retraction on the Stiefel manifold that has a closed-form inverse, improving efficiency over existing methods.

Findings

First second-order retraction with closed-form inverse on Stiefel manifold.

Efficient computation due to closed-form inverse.

Outperforms existing retractions in accuracy and invertibility.

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.