Optimization Methods for Joint Eigendecomposition
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929710156480512 |
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| author | Troedsson, Erik Carlsson, Marcus Wendt, Herwig |
| author_facet | Troedsson, Erik Carlsson, Marcus Wendt, Herwig |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07484 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimization Methods for Joint Eigendecomposition Troedsson, Erik Carlsson, Marcus Wendt, Herwig Optimization and Control 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. |
| title | Optimization Methods for Joint Eigendecomposition |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2502.07484 |