BEST NONSPHERICAL SYMMETRICLOW RANK APPROXIMATION

摘要

The symmetry preserving singular value decomposition (SPSVD) produces the bestsymmetric (low rank) approximation to a set of data. These symmetric approximations are charac-terized via an invariance under the action of a symmetry group on the set of data. The symmetrygroups of interest consist of all the nonspherical symmetry groups in three dimensions. This setincludes the rotational, reflectional, dihedral, and inversion symmetry groups. In order to calculatethe 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 formu-lated. These new matrix representations lead directly to a novel reweighting iterative method todetermine the symmetry of a given data set by solving a series of minimization problems. Once thesymmetry of the data set is found, the best symmetric (low rank) approximation in the Frobeniusnorm and matrix 2-norm can be established by using the SPSVD.