Abstract
This paper introduces and investigates a gradient flow of the log likelihood function restricted on the isospectral submanifold and proves that the flow globally converges to diagonal matrices for almost all positive definite initial matrices. This result shows that the log likelihood function does not have any spurious stable fixed point and ensures the global convergence of ICA algorithms based on the log likelihood function.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Brockett, R.W.: Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems. Linear Algebra Appl. 146, 79–91 (1991)
Brockett, R.W.: Differential geometry and the design of gradient algorithms. In: Green, R., Yau, S.-T. (eds.) Differential Geometry: Partial Differential Equations on Manifolds, pp. 69–92. Amer. Math. Soc., Providence (1993)
Chu, M.T., Driessel, K.R.: The projected gradient method for least squares matrix approximations with spectral constraints. SIAM J. Numer. Anal. 27, 1050–1060 (1990)
Flury, B.N.: Common principal components in k groups. J. Amer. Statist. Assoc. 79, 892–897 (1984)
Hori, G.: Isospectral gradient flows for non-symmetric eigenvalue problem. Japan J. Indust. Appl. Math. 17, 27–42 (2000)
Hori, G.: A new approach to joint diagonalization. In: Proc. Intl. Workshop Independent Component Analysis Blind Signal Separation, pp. 151–155 (2000)
Manton, J.H.: Optimisation algorithms exploiting unitary constraints. IEEE Trans. Signal Processing 50, 635–650 (2002)
Pham, D.T.: Joint approximate diagonalization of positive definite Hermitian matrices. SIAM J. Matrix Anal. Appl. 22, 1136–1152 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hori, G. (2006). Global Analysis of Log Likelihood Criterion. In: Rosca, J., Erdogmus, D., Príncipe, J.C., Haykin, S. (eds) Independent Component Analysis and Blind Signal Separation. ICA 2006. Lecture Notes in Computer Science, vol 3889. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11679363_101
Download citation
DOI: https://doi.org/10.1007/11679363_101
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32630-4
Online ISBN: 978-3-540-32631-1
eBook Packages: Computer ScienceComputer Science (R0)