Orthonormal approximate joint block-diagonalization Bloc-diagonalisation simultanée approchée avec contrainte orthonormale

The aim of this work is to give a comprehensive overview of the problem of jointly block-diagonalizing a set of matrices. We discuss how to implement methods in the common case of only approximative block-diagonalizability using Jacobi algorithms. Standard Jacobi optimization techniques for diagonalization and joint diagonalization are reviewed first, before we study their generalizations to the block case and give some new theoretical insights on existence and uniqueness issues as well as on the interplay between block and standard diagonalization problems. Simulations on synthetic data show that in the block case convergence to the optimal solution is not always observed in practice and that the behavior of the Jacobi approach is very much dependent on the initialization of the orthonormal basis and also on the choice of the successive rotations. Résumé: Ce rapport présente demanì ere unifiée des techniques de diago-nalisation, bloc-diagonalisation (BD), diagonalisation simultanée (DS) et bloc-diagonalisation simultanée (BDS) par méthode de Jacobi. Nos contributions principales concernent leprobì eme de la bloc-diagonalisation simultanée. Les conditions d'existence et d'unicité des solutions sontétudiées. Il apparaˆıt en pratique que la convergence des méthodes de Jacobi vers une solution opti-male (minimisant le critère choisi), généralement observée dans le cas de la DS, n'est pas toujours observée pour la BDS, et qu'elle dépend largement de l'initialisation et du choix des rotations successives. A ce titre nous décrivons une nouvelle méthode de sélection des rotations qui maximise d'un point de vue empirique les chances de convergence vers une solution optimale (sans toutefois la garantir). Abstract. The aim of this work is to give a comprehensive overview of the problem of jointly block-diagonalizing a set of matrices. We discuss how to implement methods in the common case of only approximative block-diagonalizability using Jacobi algorithms. Standard Jacobi optimization techniques for diagonalization and joint diagonalization are reviewed first, before we study their generalizations to the block case and give some new theoretical insights on existence and uniqueness issues as well as on the interplay between block and standard diagonalization problems. Simulations on synthetic data show that in the block case convergence to the optimal solution is not always observed in practice and that the behavior of the Jacobi approach is very much dependent on the initialization of the orthonormal basis and also on the choice of the successive rotations. 1. Introduction. Joint diagonalization techniques have received much attention in the last fifteen years within the field of …