A Geometric Newton Method for Oja's Vector Field.

Absil, Pierre-Antoine;Ishteva, Mariya;De Lathauwer, Lieven;Van Huffel, Sabine
(2008) Neural Computation — Vol. 21, n° 5, p. 1415-1433 (2009)

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  • Absil, Pierre-AntoineUCLouvain
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  • Ishteva, MariyaResearch Division SCD, Department of Electrical Engineering, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium
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  • De Lathauwer, LievenKULSubfaculty Science and Technology, Katholieke Universiteit Leuven, 8500 Kortrijk, Belgium, and Research Division SCD, Department of Electrical Engineering, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium
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  • Van Huffel, SabineResearch Division SCD, Department of Electrical Engineering, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium
    Author
Abstract
Newton's method for solving the matrix equation F(X) identical with AX - XX(T)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 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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Absil, P.-A., Ishteva, M., De Lathauwer, L., & Van Huffel, S. (2008). A Geometric Newton Method for Oja’s Vector Field. Neural Computation, 21(5), 1415-1433. https://doi.org/10.1162/neco.2008.04-08-749 (Original work published 2009)