CoRMF：临界性排序的循环平均场伊辛求解器
CoRMF: Criticality-Ordered Recurrent Mean Field Ising Solver
AI总结：
本研究提出基于RNN的CoRMF伊辛求解器，通过临界性排序自旋序列结合自回归平均场分解与RNN优化，实现高效正向伊辛推断，具备模块化通用性，且在理论和数值实验中均验证了其优势。
摘要：
我们提出了一种基于循环神经网络（RNN）的高效伊辛模型求解器——临界性排序循环平均场（Criticality-ordered Recurrent Mean Field，CoRMF），用于求解正向伊辛问题。该方法的核心是通过贪心算法对关键任务边进行排序，得到N自旋伊辛模型的临界性排序自旋序列，从而可以利用自回归平均场分解，并通过循环神经网络（RNN）对其进行优化。我们的方法有两个显著特点：（i）通过利用底层伊辛图的近似树结构，新得到的临界性排序实现了变分平均场与RNN的统一，使得通常难以求解的伊辛模型可以通过概率推断进行高效探究；（ii）该方法模块化程度高、与模型无关，同时具有足够的表达能力，因此只需极少工作量即可完全适用于任何正向伊辛推断问题。在计算层面，通过使用方差缩减的蒙特卡洛梯度估计器，CoRMF以自训练方式求解伊辛问题，无需数据/证据，且推断任务可通过直接从RNN采样来执行。在理论层面，我们利用矩阵割分解机制，证明了该方法比朴素平均场具有更紧的误差界。在数值层面，我们在一系列伊辛数据集上验证了该框架的实用性。
英文摘要：
We propose an RNN-based efficient Ising model solver, the Criticality-ordered Recurrent Mean Field (CoRMF), for forward Ising problems. In its core, a criticality-ordered spin sequence of an $N$-spin Ising model is introduced by sorting mission-critical edges with greedy algorithm, such that an autoregressive mean-field factorization can be utilized and optimized with Recurrent Neural Networks (RNNs). Our method has two notable characteristics: (i) by leveraging the approximated tree structure of the underlying Ising graph, the newly-obtained criticality order enables the unification between variational mean-field and RNN, allowing the generally intractable Ising model to be efficiently probed with probabilistic inference; (ii) it is well-modulized, model-independent while at the same time expressive enough, and hence fully applicable to any forward Ising inference problems with minimal effort. Computationally, by using a variance-reduced Monte Carlo gradient estimator, CoRFM solves the Ising problems in a self-train fashion without data/evidence, and the inference tasks can be executed by directly sampling from RNN. Theoretically, we establish a provably tighter error bound than naive mean-field by using the matrix cut decomposition machineries. Numerically, we demonstrate the utility of this framework on a series of Ising datasets.
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