A theory of condition for unconstrained perturbations
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Traditionally, the theory of condition numbers assumes errors in the data that are constrained to be tangent to the input space. We extend this theory to unconstrained perturbations in the case when the input space is an embedded submanifold of an ambient Euclidean space. We will show that the way how the input space is curved inside the ambient space affects the condition number. Furthermore, we exhibit a connection to the sensitivity analysis of Riemannian optimization problems. We validate our main results in two prototypical applications by numerical experiments. The applications we consider are recovery of low-rank matrices in compressed sensing and n-camera triangulation in computer vision.
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