• TITLE: A geometric Newton method for Oja's vector field
• AUTHORS: P.-A. Absil, M. Ishteva, L. De Lathauwer, S. Van Huffel
• 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.
• KEY WORDS: Oja's learning equation, Oja's flow, differential-geometric optimization, Riemannian optimization, quotient manifold, neural networks
• STATUS: Neural Computation, Vol. 21, No. 5, Pages 1415-1433, May 2009

• Neural Computation, 2009: doi:10.1162/neco.2008.04-08-749
• Tech report: http://arxiv.org/abs/0804.0989

BibTeX citation:

@ARTICLE{AbsIshLatHuf2009,
  author = "P.-A. Absil and M. Ishteva and L. De Lathauwer and S. Van Huffel",
  title = "A geometric {Newton} method for {Oja}'s vector field",
  journal = "Neural Comput.",
  fjournal = "Neural Computation",
  year = 2009,
  month = "May",
  volume = 21,
  number = 5,
  pages = "1415--1433",
}