### Browsing by Author "Afsari, Bijan"

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Item Gradient Flow Based Matrix Joint Diagonalization for Independent Componenet Analysis(2004-05-06) Afsari, Bijan; Krishnaprasad, P. S.; Electrical EngineeringIn 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 O(n) and GL(n).\\ 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.Item Gradient Flow Based Matrix Joint Diagonalization for Independent Component Analysis(2004) Afsari, Bijan; Krishnaprasad, P.S.; ISRIn 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 O(n) and GL(n). 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.Item MEANS AND AVERAGING ON RIEMANNIAN MANIFOLDS(2009) Afsari, Bijan; Krishnaprasad, P.S.; Grove, Karsten; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Processing of manifold-valued data has received considerable attention in recent years. Standard data processing methods are not adequate for such data. Among many related data processing tasks finding means or averages of manifold-valued data is a basic and important one. Although means on Riemannian manifolds have a long history, there are still many unanswered theoretical questions about them, some of which we try to answer. We focus on two classes of means: the Riemannian $L^{p}$ mean and the recursive-iterative means. The Riemannian $L^{p}$ mean is defined as the solution(s) of a minimization problem, while the recursive-iterative means are defined based on the notion of Mean-Invariance (MI) in a recursive and iterative process. We give a new existence and uniqueness result for the Riemannian $L^{p}$ mean. The significant consequence is that it shows the local and global definitions of the Riemannian $L^{p}$ mean coincide under an uncompromised condition which guarantees the uniqueness of the local mean. We also study smoothness, isometry compatibility, convexity and noise sensitivity properties of the $L^{p}$ mean. In particular, we argue that positive sectional curvature of a manifold can cause high sensitivity to noise for the $L^{2}$ mean which might lead to a non-averaging behavior of that mean. We show that the $L^{2}$ mean on a manifold of positive curvature can have an averaging property in a weak sense. We introduce the notion of MI, and study a large class of recursive-iterative means. MI means are related to an interesting class of dynamical systems that can find Riemannian convex combinations. A special class of the MI means called pairwise mean, which through an iterative scheme called Perimeter Shrinkage is related to cyclic pursuit on manifolds, is also studied. Finally, we derive results specific to the special orthogonal group and the Grassmannian manifold, as these manifolds appear naturally in many applications. We distinguish the $2$-norm Finsler balls of appropriate radius in these manifolds as domains for existence and uniqueness of the studied means. We also introduce some efficient numerical methods to perform the related calculations in the specified manifolds.Item A Novel Non-Orthogonal Joint Diagonalization Cost Function for ICA(2005) Afsari, Bijan; Krishnaprasad, Perinkulam S.; Krishnaprasad, Perinkulam S.; ISRWe present a new scale-invariant cost function for non-orthogonal joint-diagonalization of a set of symmetric matrices with application to Independent Component Analysis (ICA). We derive two gradient minimization schemes to minimize this cost function. We also consider their performance in the context of an ICA algorithm based on non-orthogonal joint diagonalization.