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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2506.20768 |
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| _version_ | 1866914415354314752 |
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| author | Yu, Zhiyuan Liu, Jingbo |
| author_facet | Yu, Zhiyuan Liu, Jingbo |
| contents | Approximate inference is central to Bayesian learning, with variational inference (VI) providing a scalable framework for posterior approximation. While mean-field VI often fails in high dimensions, the more refined Bethe approximation, equivalent to the Thouless-Anderson-Palmer (TAP) free energy in statistical physics, has long been conjectured to capture Bayes-optimal behavior. We prove that the TAP formula holds for Bayesian linear regression with a uniform spherical prior at all noise levels ($Δ>0$), extending the result of Qiu and Sen (2023) in the high-noise regime. Our argument constructs a ridge regression functional that dominates the TAP free energy, yielding the first rigorous analysis of the global optimizer of the non-concave TAP functional for a planted inference model at an arbitrary noise level. This verifies that TAP, rather than mean-field, is the correct variational description in this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of The TAP Free Energy for High-Dimensional Linear Regression with Spherical Priors at All Temperatures Yu, Zhiyuan Liu, Jingbo Statistics Theory Approximate inference is central to Bayesian learning, with variational inference (VI) providing a scalable framework for posterior approximation. While mean-field VI often fails in high dimensions, the more refined Bethe approximation, equivalent to the Thouless-Anderson-Palmer (TAP) free energy in statistical physics, has long been conjectured to capture Bayes-optimal behavior. We prove that the TAP formula holds for Bayesian linear regression with a uniform spherical prior at all noise levels ($Δ>0$), extending the result of Qiu and Sen (2023) in the high-noise regime. Our argument constructs a ridge regression functional that dominates the TAP free energy, yielding the first rigorous analysis of the global optimizer of the non-concave TAP functional for a planted inference model at an arbitrary noise level. This verifies that TAP, rather than mean-field, is the correct variational description in this setting. |
| title | Proof of The TAP Free Energy for High-Dimensional Linear Regression with Spherical Priors at All Temperatures |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2506.20768 |