A geometric Newton method for Oja's vector field

TL;DR

This paper develops a geometric Newton method for solving a matrix equation related to Oja's vector field, overcoming degeneracy issues caused by symmetry and enabling more reliable convergence.

Contribution

It introduces a differential-geometric approach to modify Newton's method, addressing symmetry-related degeneracies in solving matrix equations.

Findings

The geometric Newton method successfully finds zeros of the matrix equation.

It overcomes the degeneracy issues present in the classical Newton method.

The approach leverages differential geometry to improve convergence reliability.

Abstract

Newton's method for solving the matrix equation $F(X)\equiv AX-XX^TAX=0$ runs up against the fact that its zeros are not isolated. This is due to a symmetry of $F$ by the action of the orthogonal group. We show how differential-geometric techniques can be exploited to remove this symmetry and obtain a ``geometric'' Newton algorithm that finds the zeros of $F$. The geometric Newton method does not suffer from the degeneracy issue that stands in the way of the original Newton method.