Nonorthogonal Joint Diagonalization Free of Degenerate Solution

Abstract

The problem of approximate joint diagonalization of a set of matrices is instrumental in numerous statistical signal processing applications. For nonorthogonal joint diagonalization based on the weighted least-squares (WLS) criterion, the trivial (zero) solution can simply be avoided by adopting some constraint on the diagonalizing matrix or penalty terms. However, the resultant algorithms may converge to some undesired degenerate solutions (nonzero but singular or ill-conditioned solutions). This paper discusses and analyzes the problem of degenerate solutions in detail. To solve this problem, a novel nonleast-squares criterion for approximate nonorthogonal joint diagonalization is proposed and an efficient algorithm, called fast approximate joint diagonalization (FAJD), is developed. As compared with the existing nonorthogonal diagonalization algorithms, the new algorithm can not only avoid the trivial solution but also any degenerate solutions. Theoretical analysis shows that the FAJD algorithm has some advantages over the existing nonorthogonal diagonalization algorithms. Simulation results are presented to demonstrate the efficiency of this paper's algorithm