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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.01708 |
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| _version_ | 1866915603376242688 |
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| author | Grant, Curtis Jagannath, Aukosh Ko, Justin |
| author_facet | Grant, Curtis Jagannath, Aukosh Ko, Justin |
| contents | We develop a pseudo-likelihood theory for rank one matrix estimation problems in the high dimensional limit. We prove a variational principle for the limiting pseudo-maximum likelihood which also characterizes the performance of the corresponding pseudo-maximum likelihood estimator. We show that this variational principle is universal and depends only on four parameters determined by the corresponding null model. Through this universality, we introduce a notion of equivalence for estimation problems of this type and, in particular, show that a broad class of estimation tasks, including community detection, sparse submatrix detection, and non-linear spiked matrix models, are equivalent to spiked matrix models. As an application, we obtain a complete description of the performance of the least-squares (or ``best rank one'') estimator for any rank one matrix estimation problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudo-Maximum Likelihood Theory for High-Dimensional Rank One Inference Grant, Curtis Jagannath, Aukosh Ko, Justin Statistics Theory Probability We develop a pseudo-likelihood theory for rank one matrix estimation problems in the high dimensional limit. We prove a variational principle for the limiting pseudo-maximum likelihood which also characterizes the performance of the corresponding pseudo-maximum likelihood estimator. We show that this variational principle is universal and depends only on four parameters determined by the corresponding null model. Through this universality, we introduce a notion of equivalence for estimation problems of this type and, in particular, show that a broad class of estimation tasks, including community detection, sparse submatrix detection, and non-linear spiked matrix models, are equivalent to spiked matrix models. As an application, we obtain a complete description of the performance of the least-squares (or ``best rank one'') estimator for any rank one matrix estimation problem. |
| title | Pseudo-Maximum Likelihood Theory for High-Dimensional Rank One Inference |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2503.01708 |