论文标题
张量模型中的估计
Estimation in Tensor Ising Models
论文作者
论文摘要
$ p $ tensor ising模型是用于建模依赖二进制数据的单参数离散指数族，其中足够的统计量是$ p \ geq 2 $的多线性形式。这是对矩阵模型的自然概括，该模型提供了一个方便的数学框架，用于捕获复杂的关系数据中的高阶依赖性。在本文中，我们考虑了估计$ p $ tensor ising模型的自然参数的问题，从$ n $ nodes上的分布中进行了单个样本。我们的估计值基于最大伪可能（MPL）方法，该方法提供了一种计算有效算法，用于估算避免计算棘手的分区函数的参数。我们得出了MPL估算为$ \ sqrt n $ consistent的一般条件，也就是说，它以$ 1/\ sqrt n $收敛到真实参数。特别是，我们在$ p $ -spin Sherrington-kirkpatrick（SK）型号中显示了MPL估算值的$ \ sqrt n $ - 一致性，一般$ p $ -P $ -Suromiform-rostraphs上的Spin Systems，以及HyperGraph the HyperGraph-Graph-Graph-graphstastic Block模型（HSBM）。实际上，对于HSBM，我们将相变阈值的确切位置固定下来，该位置取决于某个均值场变异问题的阳性，因此MPL估计值为$ \ sqrt n $ constistert，而低于阈值，则MPL估计值是一致的。最后，我们在$ p $ tensor curie-weiss型号的特殊情况下得出了MPL估计的确切波动。我们结果的一个有趣结果是，Curie-Weiss模型中的MPL估计值在估计阈值以上的所有点上都饱和，即，即使仅通过使计算处理能力的真实近似值来最大程度地近似，MPL估计不会造成渐近效率的损失。
The $p$-tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree $p \geq 2$. This is a natural generalization of the matrix Ising model, that provides a convenient mathematical framework for capturing higher-order dependencies in complex relational data. In this paper, we consider the problem of estimating the natural parameter of the $p$-tensor Ising model given a single sample from the distribution on $N$ nodes. Our estimate is based on the maximum pseudo-likelihood (MPL) method, which provides a computationally efficient algorithm for estimating the parameter that avoids computing the intractable partition function. We derive general conditions under which the MPL estimate is $\sqrt N$-consistent, that is, it converges to the true parameter at rate $1/\sqrt N$. In particular, we show the $\sqrt N$-consistency of the MPL estimate in the $p$-spin Sherrington-Kirkpatrick (SK) model, spin systems on general $p$-uniform hypergraphs, and Ising models on the hypergraph stochastic block model (HSBM). In fact, for the HSBM we pin down the exact location of the phase transition threshold, which is determined by the positivity of a certain mean-field variational problem, such that above this threshold the MPL estimate is $\sqrt N$-consistent, while below the threshold no estimator is consistent. Finally, we derive the precise fluctuations of the MPL estimate in the special case of the $p$-tensor Curie-Weiss model. An interesting consequence of our results is that the MPL estimate in the Curie-Weiss model saturates the Cramer-Rao lower bound at all points above the estimation threshold, that is, the MPL estimate incurs no loss in asymptotic efficiency, even though it is obtained by minimizing only an approximation of the true likelihood function for computational tractability.
