TAP FREE ENERGY, SPIN GLASSES AND VARIATIONAL INFERENCE

• 成果类型: Article
• 署名作者: Fan, Zhou; Mei, Song; Montanari, Andrea
• 署名单位: Yale University; University of California System; University of California Berkeley; Stanford University; Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1443
• 发表日期: 2021
• 页码: 1-45
• 关键词: metastable states; weighted averages; order parameters; free convolution; RANDOM MATRICES; complexity; eigenvalues; EQUATIONS; models; number

摘要

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.