Gradient Flow Based Matrix Joint Diagonalization for Independent Component Analysis
MARYLAND UNIV COLLEGE PARK INST FOR SYSTEMS RESEARCH
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In this thesis, employing the theory of matrix Lie groups, we develop gradient based flows for the problem of Simultaneous or Joint Diagonalization JD of a set of symmetric matrices. This problem has applications in many fields especially in the field of Independent Component Analysis ICA. We consider both orthogonal and non-orthogonal JD. We view the JD problem as minimization of a common quadric cost function on a matrix group. We derive gradient based flows together with suitable discretizations for minimization of this cost function on the Riemannian manifolds of On and GLn. We use the developed JD methods to introduce a new class of ICA algorithms that sphere the data, however do not restrict the subsequent search for the un-mixing matrix to orthogonal matrices. These methods provide robust ICA algorithms in Gaussian noise by making effective use of both second and higher order statistics.
- Theoretical Mathematics
- Statistics and Probability
- Operations Research