We introduce a unified framework, formulated as general latent space models, to study complex higher-order network interactions among multiple entities. Our framework covers several popular models in recent network analysis literature, including mixture multi-layer latent space model and hypergraph latent space model. We formulate the relationship between the latent positions and the observed data via a generalized multilinear kernel as the link function. While our model enjoys decent generality, its maximum likelihood parameter estimation is also convenient via a generalized tensor decomposition procedure.We propose a novel algorithm using projected gradient descent on Grassmannians. We also develop original theoretical guarantees for our algorithm. First, we show its linear convergence under mild conditions. Second, we establish finite-sample statistical error rates of latent position estimation, determined by the signal strength, degrees of freedom and the smoothness of link function, for both general and specific latent space models. We demonstrate the effectiveness of our method on synthetic data. We also showcase the merit of our method on two real-world datasets that are conventionally described by different specific models in producing meaningful and interpretable parameter estimations and accurate link prediction. We demonstrate the effectiveness of our method on synthetic data. We also showcase the merit of our method on two real-world datasets that are conventionally described by different specific models in producing meaningful and interpretable parameter estimations and accurate link prediction.

翻译：我们引入了一个统一框架,作为一般潜伏空间模型,以研究多个实体之间复杂的高阶网络互动。我们的框架涵盖最近网络分析文献中的若干流行模型,包括混合多层潜潜伏空间模型和高深潜潜空模型。我们通过一个普遍的多线性内核来制定潜伏位置与观测数据之间的关系,作为连接功能。我们的模式虽然具有体面的普遍性,但其最大可能性参数估计也通过一个普遍的高压分解程序来方便。我们提出一种新奇特的算法,使用预测的格拉斯曼尼西亚的梯度下降。我们还为我们的算法开发了原始理论保证。首先,我们展示了它在温和条件下的线性趋同。第二,我们根据信号强度、自由度和连接功能的顺利性,为一般和具体的潜伏空间模型制定了潜在位置估计的有限抽样统计错误率。我们展示了我们的合成数据方法的有效性。我们还展示了我们的方法在两个现实世界数据集上的优点。我们用不同的具体模型来进行有意义和精确的预测。我们展示了我们的方法在合成数据上的精确性模型中所作的解释。