A Framework for Generalising the Newton Method and Other Iterative Methods from Euclidean Space to Manifolds

TL;DR

This paper introduces a general framework for extending iterative methods like Newton's method from Euclidean spaces to manifolds, ensuring convergence properties are maintained and providing new insights into their design.

Contribution

It presents a coordinate-independent framework that preserves local convergence rates when generalising Newton and similar methods to manifolds.

Findings

Framework applies to any memoryless iterative method on manifolds

Coordinate changes, not Riemannian structures, are key to lifting methods

Provides new insights into Newton method design on manifolds

Abstract

The Newton iteration is a popular method for minimising a cost function on Euclidean space. Various generalisations to cost functions defined on manifolds appear in the literature. In each case, the convergence rate of the generalised Newton iteration needed establishing from first principles. The present paper presents a framework for generalising iterative methods from Euclidean space to manifolds that ensures local convergence rates are preserved. It applies to any (memoryless) iterative method computing a coordinate independent property of a function (such as a zero or a local minimum). All possible Newton methods on manifolds are believed to come under this framework. Changes of coordinates, and not any Riemannian structure, are shown to play a natural role in lifting the Newton method to a manifold. The framework also gives new insight into the design of Newton methods in general.