Spectral Initialization for High-Dimensional Phase Retrieval with Biased Spatial Directions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929286519193600 |
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| author | Bousseyroux, Pierre Potters, Marc |
| author_facet | Bousseyroux, Pierre Potters, Marc |
| contents | We explore a spectral initialization method that plays a central role in contemporary research on signal estimation in nonconvex scenarios. In a noiseless phase retrieval framework, we precisely analyze the method's performance in the high-dimensional limit when sensing vectors follow a multivariate Gaussian distribution for two rotationally invariant models of the covariance matrix C. In the first model C is a projector on a lower dimensional space while in the second it is a Wishart matrix. Our analytical results extend the well-established case when C is the identity matrix. Our examination shows that the introduction of biased spatial directions leads to a substantial improvement in the spectral method's effectiveness, particularly when the number of measurements is less than the signal's dimension. This extension also consistently reveals a phase transition phenomenon dependent on the ratio between sample size and signal dimension. Surprisingly, both of these models share the same threshold value. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15548 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Initialization for High-Dimensional Phase Retrieval with Biased Spatial Directions Bousseyroux, Pierre Potters, Marc Disordered Systems and Neural Networks Information Retrieval Probability We explore a spectral initialization method that plays a central role in contemporary research on signal estimation in nonconvex scenarios. In a noiseless phase retrieval framework, we precisely analyze the method's performance in the high-dimensional limit when sensing vectors follow a multivariate Gaussian distribution for two rotationally invariant models of the covariance matrix C. In the first model C is a projector on a lower dimensional space while in the second it is a Wishart matrix. Our analytical results extend the well-established case when C is the identity matrix. Our examination shows that the introduction of biased spatial directions leads to a substantial improvement in the spectral method's effectiveness, particularly when the number of measurements is less than the signal's dimension. This extension also consistently reveals a phase transition phenomenon dependent on the ratio between sample size and signal dimension. Surprisingly, both of these models share the same threshold value. |
| title | Spectral Initialization for High-Dimensional Phase Retrieval with Biased Spatial Directions |
| topic | Disordered Systems and Neural Networks Information Retrieval Probability |
| url | https://arxiv.org/abs/2403.15548 |