论文标题
关于腔的收敛和Bolthausen的水龙头迭代到局部磁化
On convergence of the cavity and Bolthausen's TAP iterations to the local magnetization
论文作者
论文摘要
腔和水龙头方程是Sherrington-Kirkpatrick模型中局部磁化的非线性方程的高维系统。在开创性的工作[COMM。数学。 Phys。，325（1）：333-366，2014]，Bolthausen引入了一种迭代方案，如果该模型位于Almeida-Almeida-thou thou thou thou thou thou thou-thou-Thou-Thou-Thou-Thou-Thou-Thou-Thou thouless-thouless Trunctition系列内，则对TAP方程产生渐近解决方案。但是，目前尚不清楚这种渐近溶液是否与局部磁化相吻合。在这项工作中，在空腔方程式的推动下，我们引入了一种新的迭代计划，并建立了较弱的法律。我们表明，我们的新方案在渐近上与所谓的近似信息传递算法（Bolthausen迭代的概括）相同，该迭代的概括已被普遍适用于压缩感应，贝叶斯的推论，贝叶斯的推论等。基于此，我们确认我们的空腔迭代和Bolthausen的方案既均匀地均匀地构成了当地的Magnetization，均与当地的Magnetization相关。
The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington-Kirkpatrick model. In the seminal work [Comm. Math. Phys., 325(1):333-366, 2014], Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida-Thouless transition line. However, it was unclear if this asymptotic solution coincides with the local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called Approximate Message Passing algorithm, a generalization of Bolthausen's iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen's scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated.
