Approximate Joint Singular Value Decomposition of an Asymmetric Rectangular Matrix Set

摘要

The singular value decomposition is among the most useful and widespread tools in linear algebra. Often in engineering a multitude of matrices with common latent structure are available. Suppose we have a set of matrices for which we wish to find two orthogonal matrices and such that all products are as close as possible to rectangular diagonal form. We show that the problem can be solved efficiently by iterating either power iterations followed by an orthogonalization process or Givens rotations. The two proposed algorithms can be seen as a generalization of approximate joint diagonalization (AJD) algorithms to the bilinear orthogonal forms. Indeed, if the input matrices are symmetric and , the optimization problem reduces to that of orthogonal AJD. The effectiveness of the algorithms is shown with numerical simulations and the analysis of a large database of 84 electroencephalographic recordings. The proposed algorithms open the road to new applications of the blind source separation framework, of which we give some example for electroencephalographic data.