Spherical coordinate-based kernel principal component analysis

Abstract

This paper proposes a spherical coordinate-based kernel principal component analysis (PCA). Here, the kernel function is the nonlinear transform from the Cartesian coordinate system to the spherical coordinate system. In particular, first, the vectors represented in the Cartesian coordinate system are expressed as those represented in the spherical coordinate system. Then, certain rotational angles or the radii of the vector are set to their corresponding mean values. Finally, the processed vectors represented in the spherical coordinate system are expressed back in the Cartesian coordinate system. As the degrees of the freedoms of the processed vectors represented in the spherical coordinate system are reduced, the dimension of the manifold of the processed vectors represented in the Cartesian coordinate system is also reduced. Moreover, since the conversion between the vectors represented in the Cartesian coordinate system and those represented in the spherical coordinate system only involves some elements in the vectors, the required computational power for the conversion is low. Computer numerical simulation results show that the mean squares reconstruction error via the spherical coordinate-based kernel PCA is lower than that via the conventional PCA. Also, the required computational power is significantly reduced.