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Lagrangian Gradient for Principal Singular Component Analysis

In this paper a framework for developing dynamical systems for solving optimization problems with orthogonal constraints are proposed. These systems are based on the Lagrangian gradient of the given constrained problem. By exploiting orthogonality and symmetry in the constraints, several dynamical systems for solving the same optimization problem are developed, and conditions for global stability of these systems are also given. As a special case, the reduced singular value decomposition is formulated as an optimization problem within this framework which resulted in a singular value dynamical system whose solution converges to the principal singular components of a given matrix.