Asymptotic quadratic convergence of the Gauss-Newton method for complex phase retrieval
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913391217475584 |
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| author | Huang, Meng |
| author_facet | Huang, Meng |
| contents | In this paper, we introduce a Gauss-Newton method for solving the complex phase retrieval problem. In contrast to the real-valued setting, the Gauss-Newton matrix for complex-valued signals is rank-deficient and, thus, non-invertible. To address this, we utilize a Gauss-Newton step that moves orthogonally to certain trivial directions. We establish that this modified Gauss-Newton step has a closed-form solution, which corresponds precisely to the minimal-norm solution of the associated least squares problem. Additionally, using the leave-one-out technique, we demonstrate that $m\ge O( n\log^3 n)$ independent complex Gaussian random measurements ensures that the entire trajectory of the Gauss-Newton iterations remains confined within a specific region of incoherence and contraction with high probability. This finding allows us to establish the asymptotic quadratic convergence rate of the Gauss-Newton method without the need of sample splitting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic quadratic convergence of the Gauss-Newton method for complex phase retrieval Huang, Meng Numerical Analysis Information Theory In this paper, we introduce a Gauss-Newton method for solving the complex phase retrieval problem. In contrast to the real-valued setting, the Gauss-Newton matrix for complex-valued signals is rank-deficient and, thus, non-invertible. To address this, we utilize a Gauss-Newton step that moves orthogonally to certain trivial directions. We establish that this modified Gauss-Newton step has a closed-form solution, which corresponds precisely to the minimal-norm solution of the associated least squares problem. Additionally, using the leave-one-out technique, we demonstrate that $m\ge O( n\log^3 n)$ independent complex Gaussian random measurements ensures that the entire trajectory of the Gauss-Newton iterations remains confined within a specific region of incoherence and contraction with high probability. This finding allows us to establish the asymptotic quadratic convergence rate of the Gauss-Newton method without the need of sample splitting. |
| title | Asymptotic quadratic convergence of the Gauss-Newton method for complex phase retrieval |
| topic | Numerical Analysis Information Theory |
| url | https://arxiv.org/abs/2406.09903 |