Optimization Methods for Joint Eigendecomposition

Abstract: Joint diagonalization, the process of finding a shared set of approximate eigenvectors for a collection of matrices, arises in diverse applications such as multidimensional harmonic analysis or quantum information theory. This task is typically framed as an optimization problem: minimizing a non-convex function that quantifies off-diagonal matrix elements across possible bases. In this work, we introduce a suite of efficient algorithms designed to locate local minimizers of this functional. Our methods leverage the Hessian's structure to bypass direct computation of second-order derivatives, evaluating it as either an operator or bilinear form - a strategy that remains computationally feasible even for large-scale applications. Additionally, we demonstrate that this Hessian-based information enables precise estimation of parameters, such as step-size, in first-order optimization techniques like Gradient Descent and Conjugate Gradient, and the design of second-order methods such as (Quasi-)Newton. The resulting algorithms for joint diagonalization outperform existing techniques, and we provide comprehensive numerical evidence of their superior performance.