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

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.