Summary: 07 Apr 2008 Tech. Report UCL-INMA-2008.013
A geometric Newton method for Oja's vector field
L. De Lathauwer§
S. Van Huffel
April 7, 2008
Newton's method for solving the matrix equation F(X) AX -
XXT AX = 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 re-
move 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
Key words. Oja's learning equation, Oja's flow, differential-geometric optimization, Rie-
mannian optimization, quotient manifold, neural networks
Let A be a symmetric positive-definite n × n matrix and let p be a positive integer smaller