In this paper, we study two well known methods of Ising structure learning, namely the pseudolikelihood approach and the interaction screening approach, in the context of tensor recovery in $k$-spin Ising models. We show that both these approaches, with proper regularization, retrieve the underlying hypernetwork structure using a sample size logarithmic in the number of network nodes, and exponential in the maximum interaction strength and maximum node-degree. We also track down the exact dependence of the rate of tensor recovery on the interaction order $k$, that is allowed to grow with the number of samples and nodes, for both the approaches. Finally, we provide a comparative discussion of the performance of the two approaches based on simulation studies, which also demonstrate the exponential dependence of the tensor recovery rate on the maximum coupling strength.

翻译：本文研究了两种著名的Ising结构学习方法——伪似然方法和交互筛选方法——在$k$-自旋Ising模型的张量恢复背景下的应用。我们证明，在适当正则化下，这两种方法都能以网络节点数的对数采样量级恢复底层超网络结构，同时该采样量级与最大交互强度和最大节点度数呈指数关系。我们还精确推导了两种方法中张量恢复速率对交互阶数$k$的依赖关系，其中$k$允许随样本量和节点数增长。最后，我们基于仿真研究对两种方法的性能进行了比较讨论，研究结果同样表明张量恢复速率与最大耦合强度呈指数依赖关系。