We present a new family of zero-field Ising models over $N$ binary variables/spins obtained by consecutive "gluing" of planar and $O(1)$-sized components and subsets of at most three vertices into a tree. The polynomial-time algorithm of the dynamic programming type for solving exact inference (computing partition function) and exact sampling (generating i.i.d. samples) consists in a sequential application of an efficient (for planar) or brute-force (for $O(1)$-sized) inference and sampling to the components as a black box. To illustrate the utility of the new family of tractable graphical models, we first build a polynomial algorithm for inference and sampling of zero-field Ising models over $K_{3,3}$-minor-free topologies and over $K_{5}$-minor-free topologies -- both are extensions of the planar zero-field Ising models -- which are neither genus - nor treewidth-bounded. Second, we demonstrate empirically an improvement in the approximation quality of the NP-hard problem of inference over the square-grid Ising model in a node-dependent non-zero "magnetic" field.

中文翻译：

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平面零场 Ising 模型的易处理无次要推广

我们在 $N$ 二进制变量/自旋上提出了一个新的零场 Ising 模型系列，该模型是通过将平面和 $O(1)$ 大小的组件和最多三个顶点的子集连续“粘合”到树中而获得的。用于求解精确推理（计算分区函数）和精确采样（生成 iid 样本）的动态规划类型的多项式时间算法包括高效（对于平面）或蛮力（对于 $O(1) $-sized) 作为黑盒对组件进行推理和采样。为了说明新的易处理图形模型系列的实用性，我们首先构建了一个多项式算法，用于对 $K_{3 上的零场 Ising 模型进行推理和采样，3}$-minor-free 拓扑和 $K_{5}$-minor-free 拓扑——两者都是平面零场 Ising 模型的扩展——它们既不是属的——也不是树宽有界的。其次，我们凭经验证明了在节点相关的非零“磁场”中对方形网格 Ising 模型进行推理的 NP 难问题的近似质量的改进。