High-Order Retractions on Matrix Manifolds Using Projected Polynomials

Abstract

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.

Physics-Informed Tolerance Allocation: A Surrogate-Based Framework for the Control of Geometric Variation on System Performance

“…The mathematically ideal retraction is the Riemannian exponential map which maps a point τ ∈ M and tangent vector u ∈ T τ M to a point along a geodesic curve on the manifold M which starts at τ in the direction of u. However, the exponential map is too computationally demanding to use in practice, so several alternative retractions have been proposed in the literature [3,4,24]. A particularly convenient class of retractions is based on the concept of a retractor, which is formally defined below for our problem setting.…”