A geometric Newton method for Oja's vector field

Details A geometric Newton method for Oja's vector field A geometric Newton method for Oja's vector field

2008-04-07

arxiv.org/abs/0804.0989v1

Neural Computation, Vol. 21, No. 5, Pages 1415-1433, May 2009

Mathematics

Numerical Analysis

Scientific paper

10.1162/neco.2008.04-08-749

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.

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