The condition number of Riemannian approximation problems

TL;DR

This paper analyzes the sensitivity of Riemannian least-squares inverse problems, deriving a condition number involving curvature, and validates findings through computer vision experiments.

Contribution

It introduces a novel condition number for Riemannian inverse problems that incorporates curvature via the Weingarten map.

Findings

Derived a first-order sensitivity measure for Riemannian least-squares problems.

Connected the condition number to the curvature of the input manifold.

Validated theoretical results with experiments on camera triangulation.

Abstract

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.