Principal Component Analysis which Applies an Optimization Method on a Hyper-spherical Surface

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Principal component analysis (PCA) is a fundamental tool for processing multidimensional data. Although various algorithms for finding principal directions have been proposed, they are interpreted as optimizing a certain objective function with orthogonal constraints which is incorporated into its Lagrangean as penalty functions.In this papar, we propose a new method based on constrained optimization technique which directly transform the constrained decision variables into the unconstrained ones through nonlinear functions. This method takes advantage in numerical computations and has extensibility to the global optimization aided by chaos. We start from reformulating PCA as an optimization problem on a hyper-spherical surface with orthogonality constraints. Then, we propose a new nonlinear transformation by considering the rotational operation in Euclid's space. Finally, we derive its equivalent unconstrained optimization problem and provide the gradient dynamics for solving it analytically. The validity of our proposing methods is demonstrated with numerical simulations.
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