Abstract
We propose a class of second order mechanical systems on Grassmann manifolds that converge to the dominant eigenspace of a given symmetric matrix. Such second order flows for principal subspace analysis are derived from a modification of the familiar Euler-Lagrange equation by inserting a suitable damping term. The kinetic energy of the system is defined by the Riemannian metric on the Grassmannian while the potential energy is given by a trace function, that is defined by the symmetric matrix. Convergence of the algorithm to the dominant subspace is shown for generic initial conditions and a comparison with the Oja flow is made.
Partially supported by the German Research Foundation under grant KONNEW HE 1858/10-1
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References
F. Alvarez, H. Attouch, J. Bolte, and P. Redont. A second-order gradient-like dissipative dynamical system with hessian driven damping. J. MPA, 19:595–603, 2002.
F. Alvarez, J. Bolte, and O. Brahic. Hessian Riemannian gradient flows in convex programming. SIAM J. Control and Opt., 43:477–501, 2004.
H. Attouch and P. Redont. The second-order in time continuous Newton methods. SIAM J. Control and Opt., 19:595–603, 1981.
A. Benveniste, M. Metivier, and P. Priouret. Adaptive Algorithms and Stochastic Approximations. Springer Publ., Berlin, 1990.
A. Bloch, P.S. Krishnaprasad, J.E. Marsden, and T.S. Ratiu. Dissipation induced instabilities. Ann. Inst. H. Poincare, Analyse Nonlineare, 11:37–90, 1994.
A. Bloch, P.S. Krishnaprasad, J.E. Marsden, and T.S. Ratiu. The Euler-Poincare equations and double bracket dissipation. Comm. Math. Physics, 175:1–42, 1996.
R.W. Brockett. Differential geometry and the design of algorithms. Proc. Symp. Pure Math., 54:69–92, 1993.
R.W. Brockett. Oscillatory descent for function minimization. In B.H.M. Alber and J. Rosenthal, editors, Current and Future Directions in Applied Mathematics, pages 65–82. Birkhäuser, Boston, 1997.
K.I. Diamantaras and S. Kung. Principal Component Neural Networks. J. Wiley, New York, 1996.
U. Helmke and P.S. Krishnaprasad. Second order dynamics for principal component analysis. 3rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Preprints, Nagoya, 2006.
U. Helmke and J. Moore. Optimization and Dynamical Systems. Springer, London, 1994.
C. Lageman. Convergence of gradient-like flows. submitted to Math. Nachrichten, 2006.
E. Oja. A simplified neuron model as a principal component analyzer. J. Math. Biology, 15:267–273, 1982.
E. Oja. Neural networks, principal components, and subspaces. Int. J. Neural Systems, 1:61–68, 1989.
S. Yoshizawa, U. Helmke, and K. Starkov. Convergence analysis for principal component flows. Int. J. Appl. Math. Comput. Sci., 11:223–236, 2001.
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Helmke, U., Krishnaprasad, P. (2007). Principal Subspace Flows Via Mechanical Systems on Grassmann Manifolds. In: Allgüwer, F., et al. Lagrangian and Hamiltonian Methods for Nonlinear Control 2006. Lecture Notes in Control and Information Sciences, vol 366. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73890-9_20
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DOI: https://doi.org/10.1007/978-3-540-73890-9_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73889-3
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