A Generalisation of the Oja subspace flow

Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS06), Kyoto, Japan, July 24 - 28, 2006

Christfried Webers and Jonathan M. Manton

Recently, a novel flow for computing the eigenvectors associated with the smallest eigenvalues of a symmetric but not necessarily positive definite matrix was introduced. This meant that the eigenvectors associated with the smallest eigenvalues could be found simply by reversing the sign of the matrix.
The current paper derives a cost function and the corresponding negative gradient flow which converges to the same subspace as is spanned by the minor components. The flow is related to Oja's major subspace flow which converges for positive definite matrices. With a suitable coordinate transformation, the Oja flow can be converted into the corresponding novel minor subspace flow. But the minor subspace flow does also converge for cases where no transformation back into an Oja flow exists. Thus, the novel flow is a generalisation of the Oja subspace flow.

Keywords : Algebraic and differential geometry, Eigenvalues and eigenfunctions, Hessian matrices, Iterative methods, Matrix decomposition, Signal subspaces, Stability analysis.

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