The $k$-tensor Ising model is an exponential family on a $p$-dimensional binary hypercube for modeling dependent binary data, where the sufficient statistic consists of all $k$-fold products of the observations, and the parameter is an unknown $k$-fold tensor, designed to capture higher-order interactions between the binary variables. In this paper, we describe an approach based on a penalization technique that helps us recover the signed support of the tensor parameter with high probability, assuming that no entry of the true tensor is too close to zero. The method is based on an $\ell_1$-regularized node-wise logistic regression, that recovers the signed neighborhood of each node with high probability. Our analysis is carried out in the high-dimensional regime, that allows the dimension $p$ of the Ising model, as well as the interaction factor $k$ to potentially grow to $\infty$ with the sample size $n$. We show that if the minimum interaction strength is not too small, then consistent recovery of the entire signed support is possible if one takes $n = \Omega((k!)^8 d^3 \log \binom{p-1}{k-1})$ samples, where $d$ denotes the maximum degree of the hypernetwork in question. Our results are validated in two simulation settings, and applied on a real neurobiological dataset consisting of multi-array electro-physiological recordings from the mouse visual cortex, to model higher-order interactions between the brain regions.

翻译：$k$-张量Ising模型是$p$维二元超立方体上的指数族分布，用于建模依赖型二元数据，其充分统计量包含观测值的所有$k$阶乘积，参数为未知的$k$阶张量，旨在捕捉二元变量间的高阶交互作用。本文描述了一种基于惩罚技术的方法，可在真实张量无任何条目过于接近零的假设下，以高概率恢复张量参数的符号支撑集。该方法基于$\ell_1$正则化的逐节点逻辑回归，以高概率恢复每个节点的符号邻域。我们的分析在高维框架下进行，允许Ising模型的维度$p$以及交互因子$k$随样本量$n$增长至无穷。研究表明，若最小交互强度并非过小，则当采样量达到$n = \Omega((k!)^8 d^3 \log \binom{p-1}{k-1})$时，可实现整个符号支撑集的一致恢复，其中$d$表示所讨论超网络的最大度数。通过两个仿真实验验证了结果，并将其应用于小鼠视觉皮层多阵列电生理记录的真实神经生物学数据集，以建模脑区间的高阶交互作用。