The condition number of Riemannian approximation problems

We consider the local sensitivity of least-squares problems for inverse problems. We assume that the sets of inputs and outputs of the inverse problem have the structures of Riemannian manifolds. The problems we consider include the approximation problem of finding the nearest point on a Riemannian embedded submanifold to a given point in the ambient space. We characterize the first-order sensitivity, i.e., condition number, of local minimizers and critical points to arbitrary perturbations of the input of the least-squares problem. This condition number involves the Weingarten map of the input manifold. We validate our main results through experiments with the $n$-camera triangulation problem in computer vision.