Approximating maps into manifolds with lower curvature bounds

Abstract

Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for functions into linear spaces in such a way that we can upper bound the forward error in terms of a lower bound on the manifold's sectional curvature. Furthermore, when the sectional curvature is nonnegative, such as for compact Lie groups, the error is guaranteed to not be worse than in the linear case. We implement the algorithm in a Julia package ManiFactor.jl and apply it to two example problems.