A direct Proof for Quadratic Convergence of the Geometric Newton Method

TL;DR

This paper provides a straightforward proof of the quadratic convergence of the geometric Newton method for finding critical points on Riemannian manifolds, simplifying previous proofs that relied on charts or additional geometric tools.

Contribution

It offers a direct proof of quadratic convergence for the geometric Newton method, avoiding complex geometric computations used in prior proofs.

Findings

Proof of quadratic convergence is direct and simpler.

The method reliably finds critical points with quadratic speed.

Applicable to Riemannian manifold optimization problems.

Abstract

We consider the problem of numerically computing a critical point of a functional $J\colon M\rightarrow R$ where $M$ is a Riemannian manifold. Due to local quadratic convergence a popular choice to solve this problem is the geometric Newton method. The proofs for quadratic convergence either use computations in a chart or require additional geometric quantities such as parallel translation. In this short note we provide a direct proof for quadratic convergence.