Joint EigenValue Decomposition Algorithms Based on First-Order Taylor Expansion

In this paper, we propose a new approach to compute the Joint EigenValue Decomposition (JEVD) of real or complex matrix sets. JEVD aims to find a common basis of eigenvectors to a set of matrices. JEVD problem is encountered in many signal processing applications. In particular, recent and efficient algorithms for the Canonical Polyadic Decomposition (CPD) of multiway arrays resort to a JEVD step. The suggested method is based on multiplicative updates. It is distinguishable by the use of a first-order Taylor Expansion to compute the inverse of the updating matrix. We call this approach Joint eigenvalue Decomposition based on Taylor Expansion (JDTE). This approach is derived in two versions based on simultaneous and sequential optimization schemes respectively. Here, simultaneous optimization means that all entries of the updating matrix are simultaneously optimized at each iteration. To the best of our knowledge, such an optimization scheme had never been proposed to solve the JEVD problem in a multiplicative update procedure. Our numerical simulations show that, in many situations involving complex matrices, the proposed approach improves the eigenvectors estimation while keeping a limited computational cost. Finally, these features are highlighted in a practical context of source separation through the CPD of telecommunication signals.