Best Nonspherical Symmetric Low Rank Approximation

Abstract

The symmetry preserving singular value decomposition (SPSVD) produces the best symmetric (low rank) approximation to a set of data. These symmetric approximations are characterized via an invariance under the action of a symmetry group on the set of data. The symmetry groups of interest consist of all the nonspherical symmetry groups in three dimensions. This set includes the rotational, reflectional, dihedral, and inversion symmetry groups. In order to calculate the best symmetric (low rank) approximation, the symmetry of the data set must be determined. Therefore, matrix representations for each of the nonspherical symmetry groups have been formulated. These new matrix representations lead directly to a novel reweighting iterative method to determine the symmetry of a given data set by solving a series of minimization problems. Once the symmetry of the data set is found, the best symmetric (low rank) approximation in the Frobenius norm and matrix 2-norm can be established by using the SPSVD.