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

Abstract

Retractions are the workhorses 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. A variety of retractions is known on the Stiefel manifold, including the Riemannian exponential map, the polar factor retraction, the QR-retraction, quasi--geodesics and the Cayley retraction. The Cayley retraction is second--order accurate under the canonical metric and features a closed-form inverse. The new retraction is the first one with the corresponding features under the Euclidean metric. We present numerical experiments which illustrates the properties of the new retraction, as well as compare it to numerous of the currently available alternatives. In addition, we examine the performance of the retraction when used for interpolation and for computing a Riemannian barycenter.