  We consider the local sensitivity of least-squares formulations of inverse
problems. The sets of inputs and outputs of these problems are assumed to have
the structures of Riemannian manifolds. The problems we consider include the
approximation problem of finding the nearest point on a Riemannian embedded
submanifold from 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, which measures the amount by which the input manifold curves in its
ambient space. We validate our main results through experiments with the
$n$-camera triangulation problem in computer vision.
