Structure learning in inverse Ising problems using ? <sub>2</sub>-regularized linear estimator
Structure learning in inverse Ising problems using ? <sub>2</sub>-regularized linear estimator
复制标题
使用 ? 逆伊辛问题中的结构学习
DOI：
10.1088/1742-5468/abfa10
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Kabashima Yoshiyuki
中科院分区：
文献类型：
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作者：
Meng Xiangming;Obuchi Tomoyuki;Kabashima Yoshiyuki
在逆伊辛问题的框架下，讨论了当采用岭2-正则化线性回归时伪李克图方法的推理性能。这种设置是为了从理论上研究数据生成模型与推理模型不同的情况，即模型不匹配的情况。在教师-学生的情况下，假设教师耦合是稀疏的，分析是使用复制品和腔的方法进行的，特别侧重于教师耦合的存在/不存在是否被正确地推断或不。结果表明，尽管存在模型失配，但在热力学极限N→∞下，当自旋数N小于数据集大小M时，可以使用朴素线性回归而无需正则化来完美地识别网络结构。此外，为了访问欠定区域M< N，我们检查了N2正则化的效果，发现所有耦合估计都出现了偏差，阻止了网络结构的完美识别。然而，我们发现，偏差呈指数衰减快，从中心自旋的距离选择在pseudolikkirk方法的增长。基于这一发现，我们提出了一个两阶段的估计：在第一阶段中，岭回归被使用，估计被修剪一个相对较小的阈值;在第二阶段的朴素线性回归只进行对剩余的耦合，和由此产生的估计再次修剪另一个相对较大的阈值。该估计器与适当的正则化系数和阈值，以实现完美的识别网络结构，即使在0< M/N< 1。大量数值实验的结果支持这些发现。
The inference performance of the pseudolikelihood method is discussed in the framework of the inverse Ising problem when the ℓ 2-regularized (ridge) linear regression is adopted. This setup is introduced for theoretically investigating the situation where the data generation model is different from the inference one, namely the model mismatch situation. In the teacher-student scenario under the assumption that the teacher couplings are sparse, the analysis is conducted using the replica and cavity methods, with a special focus on whether the presence/absence of teacher couplings is correctly inferred or not. The result indicates that despite the model mismatch, one can perfectly identify the network structure using naive linear regression without regularization when the number of spins N is smaller than the dataset size M, in the thermodynamic limit N→∞. Further, to access the underdetermined region M< N, we examine the effect of the ℓ 2 regularization, and find that biases appear in all the coupling estimates, preventing the perfect identification of the network structure. We, however, find that the biases are shown to decay exponentially fast as the distance from the center spin chosen in the pseudolikelihood method grows. Based on this finding, we propose a two-stage estimator: in the first stage, the ridge regression is used and the estimates are pruned by a relatively small threshold; in the second stage the naive linear regression is conducted only on the remaining couplings, and the resultant estimates are again pruned by another relatively large threshold. This estimator with the appropriate regularization coefficient and thresholds is shown to achieve the perfect identification of the network structure even in 0< M/N< 1. Results of extensive numerical experiments support these findings.