期刊
ANNALS OF PROBABILITY
卷 49, 期 1, 页码 1-45出版社
INST MATHEMATICAL STATISTICS-IMS
DOI: 10.1214/20-AOP1443
关键词
Bayesian inference; TAP complexity; Sherrington-Kirkpatrick model; Kac-Rice formula; free probability
资金
- Hertz Foundation Fellowship
- Office of Technology Licensing Stanford Graduate Fellowship
- [NSF DMS-1613091]
- [NSF CCF-1714305]
- [NSF IIS-1741162]
- [NSF DMS-1916198]
- [ONR N00014-18-1-272Y9]
This study investigates the Sherrington-Kirkpatrick model of spin glasses with ferromagnetically biased couplings. By considering statistical physics, it is proven that the distance between local minima of the TAP free energy and the mean of the Gibbs measure vanishes in the large size limit. The proof technique involves upper bounding the expected number of critical points of the TAP free energy using the Kac-Rice formula.
We consider the Sherrington-Kirkpatrick model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the Bayesian posterior for a high-dimensional estimation problem known as Z(2) synchronization. Statistical physics suggests to compute the expectation with respect to this Gibbs measure (the posterior mean in the synchronization problem), by minimizing the so-called Thouless-Anderson-Palmer (TAP) free energy, instead of the mean field (MF) free energy. We prove that this identification is correct, provided the ferromagnetic bias is larger than a constant (i.e., the noise level is small enough in synchronization). Namely, we prove that the scaled l(2) distance between any low energy local minimizers of the TAP free energy and the mean of the Gibbs measure vanishes in the large size limit. Our proof technique is based on upper bounding the expected number of critical points of the TAP free energy using the Kac-Rice formula.
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