Latent Space Model for Higher-Order Networks and Generalized Tensor Decomposition
提出了一个统一的潜空间模型框架，用于分析多实体间的高阶网络交互，通过广义张量分解实现参数估计，并证明了算法的线性收敛性和统计误差率。
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. Supplementary materials for this article are available online.