Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912014979301376 |
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| author | Mergny, Pierre Ko, Justin Krzakala, Florent |
| author_facet | Mergny, Pierre Ko, Justin Krzakala, Florent |
| contents | We discuss the inhomogeneous spiked Wigner model, a theoretical framework recently introduced to study structured noise in various learning scenarios, through the prism of random matrix theory, with a specific focus on its spectral properties. Our primary objective is to find an optimal spectral method and to extend the celebrated \cite{BBP} (BBP) phase transition criterion -- well-known in the homogeneous case -- to our inhomogeneous, block-structured, Wigner model. We provide a thorough rigorous analysis of a transformed matrix and show that the transition for the appearance of 1) an outlier outside the bulk of the limiting spectral distribution and 2) a positive overlap between the associated eigenvector and the signal, occurs precisely at the optimal threshold, making the proposed spectral method optimal within the class of iterative methods for the inhomogeneous Wigner problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03695 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models Mergny, Pierre Ko, Justin Krzakala, Florent Machine Learning Disordered Systems and Neural Networks Probability Statistics Theory We discuss the inhomogeneous spiked Wigner model, a theoretical framework recently introduced to study structured noise in various learning scenarios, through the prism of random matrix theory, with a specific focus on its spectral properties. Our primary objective is to find an optimal spectral method and to extend the celebrated \cite{BBP} (BBP) phase transition criterion -- well-known in the homogeneous case -- to our inhomogeneous, block-structured, Wigner model. We provide a thorough rigorous analysis of a transformed matrix and show that the transition for the appearance of 1) an outlier outside the bulk of the limiting spectral distribution and 2) a positive overlap between the associated eigenvector and the signal, occurs precisely at the optimal threshold, making the proposed spectral method optimal within the class of iterative methods for the inhomogeneous Wigner problem. |
| title | Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models |
| topic | Machine Learning Disordered Systems and Neural Networks Probability Statistics Theory |
| url | https://arxiv.org/abs/2403.03695 |