Approximate Joint Matrix Triangularization

We consider the problem of approximate joint triangularization of a set of noisy jointly diagonalizable real matrices. Approximate joint triangularizers are commonly used in the estimation of the joint eigenstructure of a set of matrices, with applications in signal processing, linear algebra, and tensor decomposition. By assuming the input matrices to be perturbations of noise-free, simultaneously diagonalizable ground-truth matrices, the approximate joint triangularizers are expected to be perturbations of the exact joint triangularizers of the ground-truth matrices. We provide a priori and a posteriori perturbation bounds on the ‘distance’ between an approximate joint triangularizer and its exact counterpart. The a priori bounds are theoretical inequalities that involve functions of the ground-truth matrices and noise matrices, whereas the a posteriori bounds are given in terms of observable quantities that can be computed from the input matrices. From a practical perspective, the problem of finding the best approximate joint triangularizer of a set of noisy matrices amounts to solving a nonconvex optimization problem. We show that, under a condition on the noise level of the input matrices, it is possible to find a good initial triangularizer such that the solution obtained by any local descent-type algorithm has certain global guarantees. Finally, we discuss the application of approximate joint matrix triangularization to canonical tensor decomposition and we derive novel estimation error bounds.