Ising模型张量学习的交互筛选与伪似然方法
Interaction Screening and Pseudolikelihood Approaches for Tensor Learning in Ising Models
AI总结：
本文研究了$k$-自旋Ising模型张量恢复中的伪似然和交互筛选方法，证明其能以对数样本量恢复超网络结构，并通过模拟和CuMiDa基因数据验证了方法的有效性。
摘要：
本文在$k$-自旋Ising模型的张量恢复背景下，研究了两种著名的Ising结构学习方法，即伪似然方法和交互筛选方法。我们证明了这两种方法在适当的正则化下，能够利用与网络节点数呈对数关系、且与最大交互强度和最大节点度呈指数关系的样本量，恢复底层的超网络结构。我们还追踪了两种方法下张量恢复率对交互阶数$k$的精确依赖关系，其中$k$允许随样本数和节点数增长。随后，我们基于模拟研究对两种方法的性能进行了比较讨论，这也证明了张量恢复率对最大耦合强度的指数依赖性。最后，我们将张量恢复方法应用于取自Curated Microarray Database (CuMiDa)的基因数据，重点理解与肝细胞癌相关的重要基因。
英文摘要：
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. We then provide a comparative discussion of the performance of the two approaches based on simulation studies, which also demonstrates the exponential dependence of the tensor recovery rate on the maximum coupling strength. Our tensor recovery methods are then applied on gene data taken from the Curated Microarray Database (CuMiDa), where we focus on understanding the important genes related to hepatocellular carcinoma.
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