High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913246956486656 |
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| author | Liang, Tengyuan Sen, Subhabrata Sur, Pragya |
| author_facet | Liang, Tengyuan Sen, Subhabrata Sur, Pragya |
| contents | We study Langevin dynamics for recovering the planted signal in the spiked matrix model. We provide a "path-wise" characterization of the overlap between the output of the Langevin algorithm and the planted signal. This overlap is characterized in terms of a self-consistent system of integro-differential equations, usually referred to as the Crisanti-Horner-Sommers-Cugliandolo-Kurchan (CHSCK) equations in the spin glass literature. As a second contribution, we derive an explicit formula for the limiting overlap in terms of the signal-to-noise ratio and the injected noise in the diffusion. This uncovers a sharp phase transition -- in one regime, the limiting overlap is strictly positive, while in the other, the injected noise overcomes the signal, and the limiting overlap is zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_04476 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models Liang, Tengyuan Sen, Subhabrata Sur, Pragya Statistics Theory Machine Learning Probability We study Langevin dynamics for recovering the planted signal in the spiked matrix model. We provide a "path-wise" characterization of the overlap between the output of the Langevin algorithm and the planted signal. This overlap is characterized in terms of a self-consistent system of integro-differential equations, usually referred to as the Crisanti-Horner-Sommers-Cugliandolo-Kurchan (CHSCK) equations in the spin glass literature. As a second contribution, we derive an explicit formula for the limiting overlap in terms of the signal-to-noise ratio and the injected noise in the diffusion. This uncovers a sharp phase transition -- in one regime, the limiting overlap is strictly positive, while in the other, the injected noise overcomes the signal, and the limiting overlap is zero. |
| title | High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models |
| topic | Statistics Theory Machine Learning Probability |
| url | https://arxiv.org/abs/2204.04476 |