High-Order Retractions on Matrix Manifolds using Projected Polynomials
This provides a new method for high-order, structure-preserving approximations on matrix manifolds, which is relevant for optimization and numerical computations on these manifolds.
The authors derive a family of high-order retractions on matrix manifolds (unitary, Grassmannian, Stiefel) using projected Bessel polynomials, achieving error O(t^{2n+1}) for approximating the Riemannian exponential map.
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 $Ω$ is a skew-Hermitian matrix and $t$ is a sufficiently small scalar, then there exists a polynomial of degree $n$ in $tΩ$ (namely, a Bessel polynomial) whose polar decomposition delivers an approximation of $e^{tΩ}$ 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.