High-Order Retractions on Matrix Manifolds using Projected Polynomials

We derive a family of high-order, structure-preserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if $\Omega$ is a skew-Hermitian matrix and $t$ is a sufficiently small scalar, then there exists a polynomial of degree $n$ in $t\Omega$ (namely, a Bessel polynomial) whose polar decomposition delivers an approximation of $e^{t\Omega}$ with error $O(t^{2n+1})$. We prove this fact and then leverage it to derive high-order approximations of the Riemannian exponential map on the Grassmannian and Stiefel manifolds. Along the way, we derive related results concerning the supercloseness of the geometric and arithmetic means of unitary matrices.